Theorem birexdv2 1222

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

Hypothesis

Ref	Expression
birexdv2.1	⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ B ∧ χ)))

Assertion

Ref	Expression
birexdv2	⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Distinct variable group(s):   φ,x

³

Proof of Theorem birexdv2

Step	Hyp	Ref	Expression
1		birexdv2.1	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ B ∧ χ)))
2	1	biexdv 936	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) ↔ ∃x(x ∈ B ∧ χ)))
3		df-rex 1206	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
4		df-rex 1206	. 2 ⊢ (∃x ∈ B χ ↔ ∃x(x ∈ B ∧ χ))
5	2, 3, 4	3bitr4g 428	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  isoini 2938  nnaordex 3191  nnawordex 3192

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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