Theorem bireudva 1317

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

Hypothesis

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

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem bireudva

Step	Hyp	Ref	Expression
1		bireudva.1	. . . . 5 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
2	1	exp 291	. . . 4 ⊢ (φ → (x ∈ A → (ψ ↔ χ)))
3	2	pm5.32d 491	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
4	3	bieudv 1013	. 2 ⊢ (φ → (∃!x(x ∈ A ∧ ψ) ↔ ∃!x(x ∈ A ∧ χ)))
5		df-reu 1207	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ))
6		df-reu 1207	. 2 ⊢ (∃!x ∈ A χ ↔ ∃!x(x ∈ A ∧ χ))
7	4, 5, 6	3bitr4g 428	1 ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Syntax hints:   → wi 2   ↔ wb 127Ú  ∧ wa 196  ∃!weu 1007   ∈ wcel 1092  ∃!wreu 1203

This theorem is referenced by:  bireudv 1318  zmax 4618  zbtwnre 4619  rebtwnz 4620

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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