Theorem ax11v 1267

Description: This is a version of ax-11o 1220 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1217 for the rederivation of ax-11o 1220 from this theorem.

Assertion

Ref	Expression
ax11v	⊢ (x = y → (φ → ∀x(x = y → φ)))

Distinct variable group:   x,y

Proof of Theorem ax11v

Step	Hyp	Ref	Expression
1		ax-16 1212	. . . 4 ⊢ (∀x x = y → ((x = y → φ) → ∀x(x = y → φ)))
2		ax-1 4	. . . 4 ⊢ (φ → (x = y → φ))
3	1, 2	syl5 21	. . 3 ⊢ (∀x x = y → (φ → ∀x(x = y → φ)))
4	3	a1d 12	. 2 ⊢ (∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
5		ax-11o 1220	. 2 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
6	4, 5	pm2.61i 126	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

Syntax hints:   → wi 3  ∀wal 956   = wceq 958

This theorem is referenced by:  sb56 1268

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-16 1212  ax-11o 1220

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