Theorem cbveu 2632

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker cbveuw 2631, cbveuvw 2630 when possible. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbveu.1	⊢ Ⅎ𝑦𝜑
cbveu.2	⊢ Ⅎ𝑥𝜓
cbveu.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbveu	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb8eu 2625	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveu.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveu.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 2531	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2610	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 278	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  Ⅎwnf 1815  [wsb 2098  ∃!weu 2593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594

This theorem is used by:  cbvreu  3404  cbvreucsf  3891