Theorem axc11r 2187

Description: Same as axc11 2314 but with reversed antecedent. Note the use of ax-12 2047 (and not merely ax12v 2048). (Contributed by NM, 25-Jul-2015.)

Assertion

Ref	Expression
axc11r	|- ( A. y y = x -> ( A. x ph -> A. y ph ) )

Proof of Theorem axc11r

Step	Hyp	Ref	Expression
1		ax-12 2047	. . 3 |- ( y = x -> ( A. x ph -> A. y ( y = x -> ph ) ) )
2	1	sps 2055	. 2 |- ( A. y y = x -> ( A. x ph -> A. y ( y = x -> ph ) ) )
3		pm2.27 42	. . 3 |- ( y = x -> ( ( y = x -> ph ) -> ph ) )
4	3	al2imi 1743	. 2 |- ( A. y y = x -> ( A. y ( y = x -> ph ) -> A. y ph ) )
5	2, 4	syld 47	1 |- ( A. y y = x -> ( A. x ph -> A. y ph ) )

Syntax hints:   -> 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-12 2047

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

This theorem is referenced by:  ax12  2304  axc11n  2307  axc11nOLD  2308  axc11nOLDOLD  2309  axc11nALT  2310  axc11  2314  hbae  2315  dral1  2325  dral1ALT  2326  axpowndlem3  9421  axc11n11r  32673  bj-ax12v3ALT  32676  bj-axc11v  32747  bj-dral1v  32748  bj-hbaeb2  32805