Theorem dveeq2-o16 2185

Description: Version of dveeq2 1940 using ax-16 2144 instead of ax-17 1616. TO DO: Recover proof from older set.mm to remove use of ax-17 1616. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveeq2-o16	⊢ (¬ ∀x x = y → (z = y → ∀x z = y))

Distinct variable group:   x,z

Proof of Theorem dveeq2-o16

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax17eq 2183	. 2 ⊢ (z = w → ∀x z = w)
2		equequ2 1686	. 2 ⊢ (w = y → (z = w ↔ z = y))
3	1, 2	dvelimALT 2133	1 ⊢ (¬ ∀x x = y → (z = y → ∀x z = y))

Syntax hints:  ¬ wn 3   → 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  ax-12o 2142  ax-16 2144

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)