Theorem rcla4edv 1879

Description: Restricted existential specialization with implicit substitution. (Contributed by FL, 17-Apr-2007.)

Hypothesis

Ref	Expression
rcla4dv.1	⊢ ((φ ⋀ x = A) → (ψ ↔ χ))

Assertion

Ref	Expression
rcla4edv	⊢ ((φ ⋀ A ∈ B) → (χ → ∃x ∈ B ψ))

Distinct variable groups:   x,A   x,B   φ,x   χ,x

Proof of Theorem rcla4edv

Step	Hyp	Ref	Expression
1		rcla4dv.1	. . . . . . . 8 ⊢ ((φ ⋀ x = A) → (ψ ↔ χ))
2	1	expcom 374	. . . . . . 7 ⊢ (x = A → (φ → (ψ ↔ χ)))
3	2	pm5.74d 585	. . . . . 6 ⊢ (x = A → ((φ → ψ) ↔ (φ → χ)))
4	3	rcla4ev 1877	. . . . 5 ⊢ ((A ∈ B ⋀ (φ → χ)) → ∃x ∈ B (φ → ψ))
5		r19.37av 1761	. . . . 5 ⊢ (∃x ∈ B (φ → ψ) → (φ → ∃x ∈ B ψ))
6	4, 5	syl 10	. . . 4 ⊢ ((A ∈ B ⋀ (φ → χ)) → (φ → ∃x ∈ B ψ))
7	6	ex 373	. . 3 ⊢ (A ∈ B → ((φ → χ) → (φ → ∃x ∈ B ψ)))
8	7	pm2.86d 71	. 2 ⊢ (A ∈ B → (φ → (χ → ∃x ∈ B ψ)))
9	8	impcom 351	1 ⊢ ((φ ⋀ A ∈ B) → (χ → ∃x ∈ B ψ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 956   ∈ wcel 958  ∃wrex 1646

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-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812

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