Theorem bj-axc10v 32717

Description: Version of axc10 2252 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc10v	|- ( A. x ( x = y -> A. x ph ) -> ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-axc10v

Step	Hyp	Ref	Expression
1		ax6v 1889	. . 3 |- -. A. x -. x = y
2		con3 149	. . . 4 |- ( ( x = y -> A. x ph ) -> ( -. A. x ph -> -. x = y ) )
3	2	al2imi 1743	. . 3 |- ( A. x ( x = y -> A. x ph ) -> ( A. x -. A. x ph -> A. x -. x = y ) )
4	1, 3	mtoi 190	. 2 |- ( A. x ( x = y -> A. x ph ) -> -. A. x -. A. x ph )
5		axc7 2132	. 2 |- ( -. A. x -. A. x ph -> ph )
6	4, 5	syl 17	1 |- ( A. x ( x = y -> A. x ph ) -> ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

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

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

This theorem is referenced by: (None)