Theorem rexeq1d 1790

Description: Equality deduction for restricted existential quantifier.

Hypothesis

Ref	Expression
raleq1d.1	|- (ph -> A = B)

Assertion

Ref	Expression
rexeq1d	|- (ph -> (E.x e. A ps <-> E.x e. B ps))

Distinct variable groups:   x,A   x,B

Proof of Theorem rexeq1d

Step	Hyp	Ref	Expression
1		raleq1d.1	. 2 |- (ph -> A = B)
2		rexeq1 1787	. 2 |- (A = B -> (E.x e. A ps <-> E.x e. B ps))
3	1, 2	syl 10	1 |- (ph -> (E.x e. A ps <-> E.x e. B ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956  E.wrex 1646

This theorem is referenced by:  rexeq12d 1795  clmi2at 7091  opnfval 7857  cmsss 7997  hlcompl 8617  pjtht 9234  pjtheut 9236  pjmvalt 9238  cayleythlem 10413

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472  df-rex 1650

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