Theorem 2rexbidva 2656

Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)

Hypothesis

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

Assertion

Ref	Expression
2rexbidva	⊢ (φ → (∃x ∈ A ∃y ∈ B ψ ↔ ∃x ∈ A ∃y ∈ B χ))

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

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

Proof of Theorem 2rexbidva

Step	Hyp	Ref	Expression
1		2ralbidva.1	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
2	1	anassrs 629	. . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → (ψ ↔ χ))
3	2	rexbidva 2632	. 2 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ ↔ ∃y ∈ B χ))
4	3	rexbidva 2632	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ ↔ ∃x ∈ A ∃y ∈ B χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃wrex 2616

This proof depends on 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-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is used by: (None)