Theorem cbveu 2103

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 2092	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveu.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveu.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 1839	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2088	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 184	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1508  [wsb 1810  ∃!weu 2079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082

This theorem is referenced by:  cbvmo  2119  cbvreu  2765  cbvreucsf  3192  tz6.12f  5668  f1ompt  5798  climeu  11861