Theorem ax11v 2096

Description: This is a version of ax-11o 2141 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See Theorem ax11v2 1992 for the rederivation of ax-11o 2141 from this theorem. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11v

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 ⊢ (φ → (x = y → φ))
2		ax16 2045	. . . 4 ⊢ (∀x x = y → ((x = y → φ) → ∀x(x = y → φ)))
3	1, 2	syl5 28	. . 3 ⊢ (∀x x = y → (φ → ∀x(x = y → φ)))
4	3	a1d 22	. 2 ⊢ (∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
5		ax11o 1994	. 2 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
6	4, 5	pm2.61i 156	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  sb56  2098  exsb  2130