Theorem rexbii2 2644

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
rexbii2.1	⊢ ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ ψ))

Assertion

Ref	Expression
rexbii2	⊢ (∃x ∈ A φ ↔ ∃x ∈ B ψ)

Proof of Theorem rexbii2

Step	Hyp	Ref	Expression
1		rexbii2.1	. . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ ψ))
2	1	exbii 1582	. 2 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃x(x ∈ B ∧ ψ))
3		df-rex 2621	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4		df-rex 2621	. 2 ⊢ (∃x ∈ B ψ ↔ ∃x(x ∈ B ∧ ψ))
5	2, 3, 4	3bitr4i 268	1 ⊢ (∃x ∈ A φ ↔ ∃x ∈ B ψ)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ 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

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-rex 2621

This theorem is referenced by:  rexeqbii  2646  rexbiia  2648  rexrab  3001  rexdifsn  3844  pw1in  4165