Theorem rexbid 2122

Description: Formula-building rule for restricted existential quantifier (deduction rule).

Hypotheses

Ref	Expression
ralbid.1	|- (ph -> A.xph)
ralbid.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
rexbid	|- (ph -> (E.x e. A ps <-> E.x e. A ch))

Proof of Theorem rexbid

Step	Hyp	Ref	Expression
1		ralbid.1	. 2 |- (ph -> A.xph)
2		ralbid.2	. . 3 |- (ph -> (ps <-> ch))
3	2	adantr 425	. 2 |- ((ph /\ x e. A) -> (ps <-> ch))
4	1, 3	rexbida 2118	1 |- (ph -> (E.x e. A ps <-> E.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   e. wcel 1300  E.wrex 2106

This theorem is referenced by:  rexbidv 2124  rexbii 2128  uniiunlem 2693  euobj1 3834  tz9.13g 5775  scott0 5847  infcvgaux1i 8480  bnj1463 13550  mgmlion 14697  homcard 14893  fgsb 14921  fgsb2 14925

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-rex 2110

Copyright terms: Public domain