Theorem ax17eq 2183

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1616 considered as a metatheorem. Do not use it for later proofs - use ax-17 1616 instead, to avoid reference to the redundant axiom ax-16 2144.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax17eq	⊢ (x = y → ∀z x = y)

Distinct variable groups:   x,z   y,z

Proof of Theorem ax17eq

Step	Hyp	Ref	Expression
1		ax-12o 2142	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
2		ax-16 2144	. 2 ⊢ (∀z z = x → (x = y → ∀z x = y))
3		ax-16 2144	. 2 ⊢ (∀z z = y → (x = y → ∀z x = y))
4	1, 2, 3	pm2.61ii 157	1 ⊢ (x = y → ∀z x = y)

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-12o 2142  ax-16 2144

This theorem is referenced by:  dveeq2-o16  2185  dveeq1-o16  2188