Theorem birexa 1229

Description: Inference adding restricted existential quantifier to both sides of an equivalence.

Hypothesis

Ref	Expression
birala.1	⊢ (x ∈ A → (φ ↔ ψ))

Assertion

Ref	Expression
birexa	⊢ (∃x ∈ A φ ↔ ∃x ∈ A ψ)

Proof of Theorem birexa

Step	Hyp	Ref	Expression
1		birala.1	. . 3 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.32i 489	. 2 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	birex2 1227	1 ⊢ (∃x ∈ A φ ↔ ∃x ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  bi2rexa 1230  elrnopab 2884  f1oweOLD 2944  unbndrank 3527  pjpj0 5259  atom1d 5750

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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