Theorem cbveuw 2690

Description: Version of cbveu 2691 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Gino Giotto, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbveuw	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbveuw

Step	Hyp	Ref	Expression
1		cbveuw.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb8euv 2685	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveuw.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveuw.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbiev 2330	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2670	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 277	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1784  [wsb 2069  ∃!weu 2653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654

This theorem is referenced by:  cbvreuw  3443  tz6.12f  6694  f1ompt  6875  climeu  14912  initoeu2  17276