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Theorem cbvmowOLD 2606
Description: Obsolete version of cbvmow 2605 as of 23-May-2024. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvmowOLD.1 𝑦𝜑
cbvmowOLD.2 𝑥𝜓
cbvmowOLD.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmowOLD (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvmowOLD
StepHypRef Expression
1 cbvmowOLD.1 . . . . 5 𝑦𝜑
21sb8ev 2355 . . . 4 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
31sb8euv 2601 . . . 4 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
42, 3imbi12i 351 . . 3 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5 moeu 2585 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 moeu 2585 . . 3 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
74, 5, 63bitr4i 303 . 2 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
8 cbvmowOLD.2 . . . 4 𝑥𝜓
9 cbvmowOLD.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
108, 9sbiev 2313 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
1110mobii 2550 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
127, 11bitri 274 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1786  wnf 1790  [wsb 2071  ∃*wmo 2540  ∃!weu 2570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-11 2158  ax-12 2175
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571
This theorem is referenced by: (None)
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