Theorem cbvreuv 3411

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3391 for a version without ax-13 2404, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker cbvreuvw 3391 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 1949	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1949	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvrmov.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvreu 3408	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃!wreu 3367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clel 2839  df-reu 3370

This theorem is used by: (None)