Theorem rexbidva 2631

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)

Hypothesis

Ref	Expression
ralbidva.1	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

Ref	Expression
rexbidva	⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem rexbidva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		ralbidva.1	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	1, 2	rexbida 2629	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃wrex 2615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2620

This theorem is referenced by:  2rexbiia  2648  2rexbidva  2655  rexeqbidva  2822  phidisjnn  4615  phialllem1  4616  dfimafn  5366  funimass4  5368  fconstfv  5456  isomin  5496  f1oiso  5499