Theorem cbveu 2635

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker cbveuw 2634, cbveuvw 2633 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 2628	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveu.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveu.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 2534	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2613	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 278	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1813  [wsb 2096  ∃!weu 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597

This theorem is referenced by:  cbvreu  3408  cbvreucsf  3897