Theorem cbvreuv 2838

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x,A   y,A   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem cbvreuv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎyφ
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		cbvralv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvreu 2834	1 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∃!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-reu 2622

This theorem is referenced by:  reu8  3033