Theorem moi2 1924

Description: Consequence of "at most one."

Hypothesis

Ref	Expression
moi2.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
moi2	⊢ (((A ∈ B ⋀ ∃*xφ) ⋀ (φ ⋀ ψ)) → x = A)

Distinct variable groups:   x,A   ψ,x

Proof of Theorem moi2

Step	Hyp	Ref	Expression
1		visset 1813	. . . . . . . . 9 ⊢ y ∈ V
2	1	eqvinc 1883	. . . . . . . 8 ⊢ (y = A ↔ ∃x(x = y ⋀ x = A))
3		hbs1 1332	. . . . . . . . . 10 ⊢ ([y / x]φ → ∀x[y / x]φ)
4		ax-17 971	. . . . . . . . . 10 ⊢ (ψ → ∀xψ)
5	3, 4	hbbi 1010	. . . . . . . . 9 ⊢ (([y / x]φ ↔ ψ) → ∀x([y / x]φ ↔ ψ))
6		sbequ12 1181	. . . . . . . . . . 11 ⊢ (x = y → (φ ↔ [y / x]φ))
7	6	bicomd 521	. . . . . . . . . 10 ⊢ (x = y → ([y / x]φ ↔ φ))
8		moi2.1	. . . . . . . . . 10 ⊢ (x = A → (φ ↔ ψ))
9	7, 8	sylan9bb 540	. . . . . . . . 9 ⊢ ((x = y ⋀ x = A) → ([y / x]φ ↔ ψ))
10	5, 9	19.23ai 1064	. . . . . . . 8 ⊢ (∃x(x = y ⋀ x = A) → ([y / x]φ ↔ ψ))
11	2, 10	sylbi 199	. . . . . . 7 ⊢ (y = A → ([y / x]φ ↔ ψ))
12	11	anbi2d 616	. . . . . 6 ⊢ (y = A → ((φ ⋀ [y / x]φ) ↔ (φ ⋀ ψ)))
13		eqeq2 1484	. . . . . 6 ⊢ (y = A → (x = y ↔ x = A))
14	12, 13	imbi12d 626	. . . . 5 ⊢ (y = A → (((φ ⋀ [y / x]φ) → x = y) ↔ ((φ ⋀ ψ) → x = A)))
15	14	cla4gv 1862	. . . 4 ⊢ (A ∈ B → (∀y((φ ⋀ [y / x]φ) → x = y) → ((φ ⋀ ψ) → x = A)))
16	15	a4sd 985	. . 3 ⊢ (A ∈ B → (∀x∀y((φ ⋀ [y / x]φ) → x = y) → ((φ ⋀ ψ) → x = A)))
17	3, 6	mo4f 1402	. . 3 ⊢ (∃*xφ ↔ ∀x∀y((φ ⋀ [y / x]φ) → x = y))
18	16, 17	syl5ib 206	. 2 ⊢ (A ∈ B → (∃*xφ → ((φ ⋀ ψ) → x = A)))
19	18	imp31 362	1 ⊢ (((A ∈ B ⋀ ∃*xφ) ⋀ (φ ⋀ ψ)) → x = A)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 954   = wceq 956   ∈ wcel 958  ∃wex 980  [wsbc 1170  ∃*wmo 1381

This theorem is referenced by:  spwpr3OLD 8662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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