Theorem cbvreuv 1335

Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution.

Hypothesis

Ref	Expression
cbvralv.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvreuv	⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)

Distinct variable group(s):   φ,y   ψ,x   x,y,A

Proof of Theorem cbvreuv

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ ((x ∈ A ∧ φ) → ∀y(x ∈ A ∧ φ))
2		ax-17 925	. . 3 ⊢ ((y ∈ A ∧ ψ) → ∀x(y ∈ A ∧ ψ))
3		eleq1 1149	. . . 4 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
4		cbvralv.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
5	3, 4	anbi12d 476	. . 3 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ ψ)))
6	1, 2, 5	cbveu 1018	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!y(y ∈ A ∧ ψ))
7		df-reu 1207	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
8		df-reu 1207	. 2 ⊢ (∃!y ∈ A ψ ↔ ∃!y(y ∈ A ∧ ψ))
9	6, 7, 8	3bitr4 158	1 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797  ∃!weu 1007   ∈ wcel 1092  ∃!wreu 1203

This theorem is referenced by:  aceq1 3552  aceq2 3554  uzwo3 4616

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-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-cleq 1097  df-clel 1099  df-reu 1207

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