Theorem rexeqbidva 2823

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Hypotheses

Ref	Expression
raleqbidva.1	⊢ (φ → A = B)
raleqbidva.2	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

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

Distinct variable groups:   x,A   x,B   φ,x

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

Proof of Theorem rexeqbidva

Step	Hyp	Ref	Expression
1		raleqbidva.2	. . 3 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
2	1	rexbidva 2632	. 2 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))
3		raleqbidva.1	. . 3 ⊢ (φ → A = B)
4	3	rexeqdv 2815	. 2 ⊢ (φ → (∃x ∈ A χ ↔ ∃x ∈ B χ))
5	2, 4	bitrd 244	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by: (None)