Theorem bi2rexdva 1234

Description: Formula-building rule for restricted existential quantifier (deduction rule).

Hypothesis

Ref	Expression
bi2raldva.1	⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))

Assertion

Ref	Expression
bi2rexdva	⊢ (φ → (∃x ∈ A ∃y ∈ B ψ ↔ ∃x ∈ A ∃y ∈ B χ))

Distinct variable group(s):   x,y,φ   y,A

Proof of Theorem bi2rexdva

Step	Hyp	Ref	Expression
1		bi2raldva.1	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ ↔ χ))
2	1	anassrs 338	. . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → (ψ ↔ χ))
3	2	birexdva 1216	. 2 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ ↔ ∃y ∈ B χ))
4	3	birexdva 1216	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ ↔ ∃x ∈ A ∃y ∈ B χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  shscomt 5284

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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