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Theorem cbvrmo 3200
 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 3198 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
51, 2, 3cbvreu 3199 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
64, 5imbi12i 339 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
7 rmo5 3192 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
8 rmo5 3192 . 2 (∃*𝑦𝐴 𝜓 ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
96, 7, 83bitr4i 292 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  Ⅎwnf 1748  ∃wrex 2942  ∃!wreu 2943  ∃*wrmo 2944 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949 This theorem is referenced by:  cbvrmov  3204  cbvdisj  4662
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