Theorem 19.42h 1577

Description: Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1578 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.42h.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
19.42h	|- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Proof of Theorem 19.42h

Step	Hyp	Ref	Expression
1		19.42h.1	. . 3 |- ( ph -> A. x ph )
2	1	19.41h 1575	. 2 |- ( E. x ( ps /\ ph ) <-> ( E. x ps /\ ph ) )
3		exancom 1499	. 2 |- ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
4		ancom 253	. 2 |- ( ( ph /\ E. x ps ) <-> ( E. x ps /\ ph ) )
5	2, 3, 4	3bitr4i 201	1 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.42v  1786