Theorem ax11 1223

Description: Rederivation of axiom ax-11 971 from the orginal version, ax-11o 1222. See theorem ax11o 1221 for the derivation of ax-11o 1222 from ax-11 971.

This theorem should not be referenced in any proof. Instead, use ax-11 971 above so that uses of ax-11 971 can be more easily identified.

Assertion

Ref	Expression
ax11	⊢ (x = y → (∀yφ → ∀x(x = y → φ)))

Proof of Theorem ax11

Step	Hyp	Ref	Expression
1		pm4.2d 171	. . . . 5 ⊢ (∀x x = y → (φ ↔ φ))
2	1	dral1 1158	. . . 4 ⊢ (∀x x = y → (∀xφ ↔ ∀yφ))
3		ax-1 4	. . . . 5 ⊢ (φ → (x = y → φ))
4	3	19.20i 996	. . . 4 ⊢ (∀xφ → ∀x(x = y → φ))
5	2, 4	syl6bir 215	. . 3 ⊢ (∀x x = y → (∀yφ → ∀x(x = y → φ)))
6	5	a1d 12	. 2 ⊢ (∀x x = y → (x = y → (∀yφ → ∀x(x = y → φ))))
7		ax-11o 1222	. . 3 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
8		ax-4 977	. . 3 ⊢ (∀yφ → φ)
9	7, 8	syl7 23	. 2 ⊢ (¬ ∀x x = y → (x = y → (∀yφ → ∀x(x = y → φ))))
10	6, 9	pm2.61i 126	1 ⊢ (x = y → (∀yφ → ∀x(x = y → φ)))

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 958   = wceq 960

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-10 970  ax-12 972  ax-4 977  ax-5o 979  ax-10o 1144  ax-11o 1222

This theorem depends on definitions:  df-bi 147  df-an 225

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