Theorem rexbidv2 2638

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)

Hypothesis

Ref	Expression
rexbidv2.1	⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ B ∧ χ)))

Assertion

Ref	Expression
rexbidv2	⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Distinct variable group:   φ,x

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

Proof of Theorem rexbidv2

Step	Hyp	Ref	Expression
1		rexbidv2.1	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ B ∧ χ)))
2	1	exbidv 1626	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) ↔ ∃x(x ∈ B ∧ χ)))
3		df-rex 2621	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
4		df-rex 2621	. 2 ⊢ (∃x ∈ B χ ↔ ∃x(x ∈ B ∧ χ))
5	2, 3, 4	3bitr4g 279	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Syntax hints:   → wi 4   ↔ 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  ax-17 1616

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

This theorem is referenced by:  rexss  3334  isoini  5498