Theorem ax-12 1402

Description: Rederive the original version of the axiom from ax-i12 1398. (Contributed by Mario Carneiro, 3-Feb-2015.)

Assertion

Ref	Expression
ax-12	|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Proof of Theorem ax-12

Step	Hyp	Ref	Expression
1		ax-i12 1398	. . . 4 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
2	1	ori 642	. . 3 |- ( -. A. z z = x -> ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
3	2	ord 643	. 2 |- ( -. A. z z = x -> ( -. A. z z = y -> A. z ( x = y -> A. z x = y ) ) )
4		ax-4 1400	. 2 |- ( A. z ( x = y -> A. z x = y ) -> ( x = y -> A. z x = y ) )
5	3, 4	syl6 29	1 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 629   A.wal 1241   = wceq 1243

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-i12 1398  ax-4 1400

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)