Theorem bireudv 1318

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

Hypothesis

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

Assertion

Ref	Expression
bireudv	⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Distinct variable group(s):   φ,x

Proof of Theorem bireudv

Step	Hyp	Ref	Expression
1		bireudv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	adantr 306	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	2	bireudva 1317	1 ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  ∃!wreu 1203

This theorem is referenced by:  reueqd 1329  oawordeu 3157  aceq6b 3565

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-eu 1009  df-reu 1207

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