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Theorem cbvrmo 3428
Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvrmow 3408, cbvrmovw 3402 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvrmo.1 𝑦𝜑
cbvrmo.2 𝑥𝜓
cbvrmo.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 cbvrmo.1 . . . 4 𝑦𝜑
2 cbvrmo.2 . . . 4 𝑥𝜓
3 cbvrmo.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrex 3362 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
51, 2, 3cbvreu 3427 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
64, 5imbi12i 350 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
7 rmo5 3399 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
8 rmo5 3399 . 2 (∃*𝑦𝐴 𝜓 ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
96, 7, 83bitr4i 303 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wnf 1782  wrex 3069  ∃!wreu 3377  ∃*wrmo 3378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380
This theorem is referenced by:  cbvrmov  3429
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