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Theorem cbvreuvw 2709
Description: Version of cbvreuv 2705 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Hypothesis
Ref Expression
cbvralvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuvw (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuvw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2238 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvralvw.1 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 473 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
4 equequ1 1712 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
53, 4bibi12d 235 . . . . 5 (𝑥 = 𝑦 → (((𝑥𝐴𝜑) ↔ 𝑥 = 𝑧) ↔ ((𝑦𝐴𝜓) ↔ 𝑦 = 𝑧)))
65cbvalvw 1919 . . . 4 (∀𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑧) ↔ ∀𝑦((𝑦𝐴𝜓) ↔ 𝑦 = 𝑧))
76exbii 1605 . . 3 (∃𝑧𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑧) ↔ ∃𝑧𝑦((𝑦𝐴𝜓) ↔ 𝑦 = 𝑧))
8 df-eu 2029 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃𝑧𝑥((𝑥𝐴𝜑) ↔ 𝑥 = 𝑧))
9 df-eu 2029 . . 3 (∃!𝑦(𝑦𝐴𝜓) ↔ ∃𝑧𝑦((𝑦𝐴𝜓) ↔ 𝑦 = 𝑧))
107, 8, 93bitr4ri 213 . 2 (∃!𝑦(𝑦𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜑))
11 df-reu 2462 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
12 df-reu 2462 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
1310, 11, 123bitr4ri 213 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wex 1492  ∃!weu 2026  wcel 2148  ∃!wreu 2457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-eu 2029  df-clel 2173  df-reu 2462
This theorem is referenced by: (None)
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