Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvrmowOLD Structured version   Visualization version   GIF version

Theorem cbvrmowOLD 3419
 Description: Obsolete version of cbvrmow 3418 as of 23-May-2024. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvrmowOLD.1 𝑦𝜑
cbvrmowOLD.2 𝑥𝜓
cbvrmowOLD.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmowOLD (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvrmowOLD
StepHypRef Expression
1 cbvrmowOLD.1 . . . 4 𝑦𝜑
2 cbvrmowOLD.2 . . . 4 𝑥𝜓
3 cbvrmowOLD.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrexw 3416 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
51, 2, 3cbvreuw 3417 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
64, 5imbi12i 354 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
7 rmo5 3407 . 2 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
8 rmo5 3407 . 2 (∃*𝑦𝐴 𝜓 ↔ (∃𝑦𝐴 𝜓 → ∃!𝑦𝐴 𝜓))
96, 7, 83bitr4i 306 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1785  ∃wrex 3131  ∃!wreu 3132  ∃*wrmo 3133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator