Theorem reubidva 1782

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

Hypothesis

Ref	Expression
reubidva.1	⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))

Assertion

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

Distinct variable group:   φ,x

Proof of Theorem reubidva

Step	Hyp	Ref	Expression
1		reubidva.1	. . . 4 ⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))
2	1	pm5.32da 651	. . 3 ⊢ (φ → ((x ∈ A ⋀ ψ) ↔ (x ∈ A ⋀ χ)))
3	2	eubidv 1388	. 2 ⊢ (φ → (∃!x(x ∈ A ⋀ ψ) ↔ ∃!x(x ∈ A ⋀ χ)))
4		df-reu 1654	. 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ⋀ ψ))
5		df-reu 1654	. 2 ⊢ (∃!x ∈ A χ ↔ ∃!x(x ∈ A ⋀ χ))
6	3, 4, 5	3bitr4g 557	1 ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 960  ∃!weu 1382  ∃!wreu 1650

This theorem is referenced by:  reubidv 1783  exfo 3828  f1ofveu 3888  zmax 6222  zbtwnre 6223  rebtwnz 6224

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-eu 1384  df-reu 1654

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