Theorem ax12 1512

Description: Rederive the original version of the axiom from ax-i12 1507. (Contributed by Mario Carneiro, 3-Feb-2015.)

Assertion

Ref	Expression
ax12	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Proof of Theorem ax12

Step	Hyp	Ref	Expression
1		ax12or 1508	. . . 4 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
2	1	ori 723	. . 3 |- ( -. A. z z = x -> ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
3	2	ord 724	. 2 |- ( -. A. z z = x -> ( -. A. z z = y -> A. z ( x = y -> A. z x = y ) ) )
4		ax-4 1510	. 2 |- ( A. z ( x = y -> A. z x = y ) -> ( x = y -> A. z x = y ) )
5	3, 4	syl6 33	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708   A.wal 1351   = wceq 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-i12 1507  ax-4 1510

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)