Theorem r19.29a 2649

Description: A commonly used pattern based on r19.29 2643. (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 1551	. 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 2647	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   E.wrex 2485

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-ral 2489  df-rex 2490

This theorem is referenced by:  cnegexlem3  8249  cnegex  8250  modqmuladdnn0  10513  uzwodc  12358  1arith  12690  mhmid  13451  mhmmnd  13452  ghmgrp  13454  ghmcmn  13663  ringinvnz1ne0  13811  neitx  14740