Theorem rexeqd 1795

Description: Equality deduction for restricted existential quantifier.

Hypothesis

Ref	Expression
raleqd.1	⊢ (A = B → (φ ↔ ψ))

Assertion

Ref	Expression
rexeqd	⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B ψ))

Distinct variable groups:   x,A   x,B

Proof of Theorem rexeqd

Step	Hyp	Ref	Expression
1		rexeq1 1790	. 2 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
2		raleqd.1	. . 3 ⊢ (A = B → (φ ↔ ψ))
3	2	rexbidv 1667	. 2 ⊢ (A = B → (∃x ∈ B φ ↔ ∃x ∈ B ψ))
4	1, 3	bitrd 530	1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B ψ))

Syntax hints:   → wi 3   ↔ wb 146   = wceq 958  ∃wrex 1649

This theorem is referenced by:  fri 2924  frc 2926  isofrlem 3907  f1oweALT 3912  zfregcl 4604  ishaus 7780  isgrp 8038  spwval 8655  spwnex 8657  pjtht 9229

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-cleq 1472  df-clel 1475  df-rex 1653

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