Theorem bj-axc10 32707

Description: Alternate (shorter) proof of axc10 2252. One can prove a version with DV(x,y) without ax-13 2246, by using ax6ev 1890 instead of ax6e 2250. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc10	|- ( A. x ( x = y -> A. x ph ) -> ph )

Proof of Theorem bj-axc10

Step	Hyp	Ref	Expression
1		ax6e 2250	. . 3 |- E. x x = y
2		exim 1761	. . 3 |- ( A. x ( x = y -> A. x ph ) -> ( E. x x = y -> E. x A. x ph ) )
3	1, 2	mpi 20	. 2 |- ( A. x ( x = y -> A. x ph ) -> E. x A. x ph )
4		axc7e 2133	. 2 |- ( E. x A. x ph -> ph )
5	3, 4	syl 17	1 |- ( A. x ( x = y -> A. x ph ) -> ph )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)