Theorem cbveu 2104

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 2093	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveu.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveu.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 1840	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2089	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 184	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1509  [wsb 1811  ∃!weu 2080

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083

This theorem is referenced by:  cbvmo  2120  cbvreu  2775  cbvreucsf  3202  tz6.12f  5698  f1ompt  5827  climeu  11977