Theorem cbvmo 2654

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 4-Jan-2023.)

Hypotheses

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

Assertion

Ref	Expression
cbvmo	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Proof of Theorem cbvmo

Step	Hyp	Ref	Expression
1		cbvmo.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb8mo 2653	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
3		cbvmo.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbvmo.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 2525	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	mobii 2594	. 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
7	2, 6	bitri 267	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 198  Ⅎwnf 1879  [wsb 2064  ∃*wmo 2589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609

This theorem is referenced by:  cbveuALT  2656  dffun6f  6114  opabiotafun  6483  2ndcdisj  21585  cbvdisjf  29895  phpreu  33875