Theorem cbvmow 2687

Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo 2688 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvmow	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvmow

Step	Hyp	Ref	Expression
1		cbvmow.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	sb8ev 2373	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3	1	sb8euv 2684	. . . 4 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
4	2, 3	imbi12i 353	. . 3 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5		moeu 2667	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6		moeu 2667	. . 3 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
7	4, 5, 6	3bitr4i 305	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
8		cbvmow.2	. . . 4 ⊢ Ⅎ𝑥𝜓
9		cbvmow.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
10	8, 9	sbiev 2329	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
11	10	mobii 2630	. 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
12	7, 11	bitri 277	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1779  Ⅎwnf 1783  [wsb 2068  ∃*wmo 2619  ∃!weu 2652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653

This theorem is referenced by:  dffun6f  6366  opabiotafun  6741  2ndcdisj  22060  cbvdisjf  30321  phpreu  34914