Theorem ax11a 926

Description: This is a version of ax-11 801 when the variables are distinct. Axiom (C8) of [Monk2] p. 105.

Assertion

Ref	Expression
ax11a	⊢ (x = y → (φ → ∀x(x = y → φ)))

Distinct variable group(s):   x,y

Proof of Theorem ax11a

Step	Hyp	Ref	Expression
1		ax-16 922	. . . 4 ⊢ (∀x x = y → ((x = y → φ) → ∀x(x = y → φ)))
2		ax-1 3	. . . 4 ⊢ (φ → (x = y → φ))
3	1, 2	syl5 22	. . 3 ⊢ (∀x x = y → (φ → ∀x(x = y → φ)))
4	3	a1d 14	. 2 ⊢ (∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
5		ax-11 801	. 2 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
6	4, 5	pm2.61i 110	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

Syntax hints:   → wi 2  ∀wal 672   = weq 797

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-11 801  ax-16 922

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