Theorem cbvreuv 3418

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3398 for a version without ax-13 2410, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2410. Use the weaker cbvreuvw 3398 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvrmov.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 1941	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1941	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvrmov.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvreu 3415	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∃!wreu 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clel 2844  df-reu 3377

This theorem is referenced by: (None)