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Theorem cbvrmo 2589
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 2587 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
51, 2, 3cbvreu 2588 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
64, 5imbi12i 237 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
7 rmo5 2582 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
8 rmo5 2582 . 2 (∃*𝑦𝐴 𝜓 ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
96, 7, 83bitr4i 210 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1394  wrex 2360  ∃!wreu 2361  ∃*wrmo 2362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-rmo 2367
This theorem is referenced by:  cbvrmov  2593  cbvdisj  3832
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