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Theorem cbvmow 2682
Description: Version of cbvmo 2683 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvmow.1 𝑦𝜑
cbvmow.2 𝑥𝜓
cbvmow.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmow (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvmow
StepHypRef Expression
1 cbvmow.1 . . . . 5 𝑦𝜑
21sb8ev 2367 . . . 4 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
31sb8euv 2679 . . . 4 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
42, 3imbi12i 353 . . 3 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5 moeu 2662 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 moeu 2662 . . 3 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
74, 5, 63bitr4i 305 . 2 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
8 cbvmow.2 . . . 4 𝑥𝜓
9 cbvmow.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
108, 9sbiev 2323 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
1110mobii 2625 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
127, 11bitri 277 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wex 1773  wnf 1777  [wsb 2062  ∃*wmo 2614  ∃!weu 2647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648
This theorem is referenced by:  dffun6f  6362  opabiotafun  6737  2ndcdisj  22056  cbvdisjf  30313  phpreu  34863
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