Theorem cbveuw 2606

Description: Version of cbveu 2608 with a disjoint variable condition, which does not require ax-10 2143, ax-13 2371. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 23-May-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	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbveuw.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbveuw.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvexv1 2343	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
5	1, 2, 3	cbvmow 2602	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
6	4, 5	anbi12i 630	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7		df-eu 2568	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
8		df-eu 2568	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1787  Ⅎwnf 1791  ∃*wmo 2537  ∃!weu 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-mo 2539  df-eu 2568

This theorem is referenced by:  cbvreuw  3341  tz6.12f  6719  f1ompt  6906  climeu  15081  initoeu2  17476