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Theorem cbvrmo 2695
Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
cbvral.1 𝑦𝜑
cbvral.2 𝑥𝜓
cbvral.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmo (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvrmo
StepHypRef Expression
1 cbvral.1 . . . 4 𝑦𝜑
2 cbvral.2 . . . 4 𝑥𝜓
3 cbvral.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrex 2693 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
51, 2, 3cbvreu 2694 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
64, 5imbi12i 238 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
7 rmo5 2685 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
8 rmo5 2685 . 2 (∃*𝑦𝐴 𝜓 ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
96, 7, 83bitr4i 211 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1453  wrex 2449  ∃!wreu 2450  ∃*wrmo 2451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-rmo 2456
This theorem is referenced by:  cbvrmov  2699  cbvdisj  3976
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