Theorem 2rexbidv 2658

Description: Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

Hypothesis

Ref	Expression
2ralbidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

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

Distinct variable groups:   φ,x   φ,y

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

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	rexbidv 2636	. 2 ⊢ (φ → (∃y ∈ B ψ ↔ ∃y ∈ B χ))
3	2	rexbidv 2636	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ ↔ ∃x ∈ A ∃y ∈ B χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∃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:  eladdc  4399  f1oiso  5500  ovelrn  5609  mucex  6134