Theorem rexbidv2 1669

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

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

Proof of Theorem rexbidv2

Step	Hyp	Ref	Expression
1		rexbidv2.1	. . 3 ⊢ (φ → ((x ∈ A ⋀ ψ) ↔ (x ∈ B ⋀ χ)))
2	1	exbidv 1281	. 2 ⊢ (φ → (∃x(x ∈ A ⋀ ψ) ↔ ∃x(x ∈ B ⋀ χ)))
3		df-rex 1653	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ⋀ ψ))
4		df-rex 1653	. 2 ⊢ (∃x ∈ B χ ↔ ∃x(x ∈ B ⋀ χ))
5	2, 3, 4	3bitr4g 557	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 960  ∃wex 982  ∃wrex 1649

This theorem is referenced by:  isoini 3906  nnaordex 4255  nnawordex 4256  rexuz 6445

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-rex 1653

Copyright terms: Public domain