Theorem r19.29a 2640

Description: A commonly used pattern based on r19.29 2634. (Contributed by Thierry Arnoux, 22-Nov-2017.)

Hypotheses

Ref	Expression
r19.29a.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
r19.29a.2	|- ( ph -> E. x e. A ps )

Assertion

Ref	Expression
r19.29a	|- ( ph -> ch )

Distinct variable groups:   ch, x   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem r19.29a

Step	Hyp	Ref	Expression
1		nfv 1542	. 2 |- F/ x ph
2		r19.29a.1	. 2 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
3		r19.29a.2	. 2 |- ( ph -> E. x e. A ps )
4	1, 2, 3	r19.29af 2638	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by:  cnegexlem3  8220  cnegex  8221  modqmuladdnn0  10477  uzwodc  12229  1arith  12561  mhmid  13321  mhmmnd  13322  ghmgrp  13324  ghmcmn  13533  ringinvnz1ne0  13681  neitx  14588