Theorem axi12 1507

Description: Proof that ax-i12 1500 follows from ax-bndl 1502. So that we can track which theorems rely on ax-bndl 1502, proofs should reference ax12or 1501 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)

Assertion

Ref	Expression
axi12	|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )

Proof of Theorem axi12

Step	Hyp	Ref	Expression
1		ax-bndl 1502	. 2 |- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )
2		sp 1504	. . . 4 |- ( A. x A. z ( x = y -> A. z x = y ) -> A. z ( x = y -> A. z x = y ) )
3	2	orim2i 756	. . 3 |- ( ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) -> ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
4	3	orim2i 756	. 2 |- ( ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) ) -> ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) ) )
5	1, 4	ax-mp 5	1 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )

Syntax hints:   -> wi 4   \/ wo 703   A.wal 1346   = wceq 1348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-bndl 1502  ax-4 1503

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)