Theorem axc10 2252

Description: Show that the original axiom ax-c10 34171 can be derived from ax6 2251 and axc7 2132 (on top of propositional calculus, ax-gen 1722, and ax-4 1737). See ax6fromc10 34181 for the rederivation of ax6 2251 from ax-c10 34171.

Normally, axc10 2252 should be used rather than ax-c10 34171, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem axc10

Step	Hyp	Ref	Expression
1		ax6 2251	. . 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  ax-13 2246

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

This theorem is referenced by: (None)