Theorem cbveu 2643

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

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 2641	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveu.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveu.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 2545	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2629	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 264	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 196  Ⅎwnf 1857  [wsb 2046  ∃!weu 2607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611

This theorem is referenced by:  cbvmo  2644  cbvreu  3308  cbvreucsf  3708  tz6.12f  6373  f1ompt  6545  climeu  14485  initoeu2  16867