Theorem cbvreuv 3438

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3412 for a version without ax-13 2380, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker cbvreuvw 3412 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 1913	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1913	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvrmov.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvreu 3435	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∃!wreu 3386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clel 2819  df-reu 3389

This theorem is referenced by: (None)