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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 𝑦𝜑
cbvmo.2 𝑥𝜓
cbvmo.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmo (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Proof of Theorem cbvmo
StepHypRef Expression
1 cbvmo.1 . . 3 𝑦𝜑
21sb8mo 2653 . 2 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
3 cbvmo.2 . . . 4 𝑥𝜓
4 cbvmo.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2525 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65mobii 2594 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
72, 6bitri 267 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wnf 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
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