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

Theorem rspcebdv 3603
 Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)
Hypotheses
Ref Expression
rspcdv.1 (𝜑𝐴𝐵)
rspcdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
rspcebdv.1 ((𝜑𝜓) → 𝑥 = 𝐴)
Assertion
Ref Expression
rspcebdv (𝜑 → (∃𝑥𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcebdv
StepHypRef Expression
1 rspcebdv.1 . . . . . . 7 ((𝜑𝜓) → 𝑥 = 𝐴)
2 rspcdv.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
31, 2syldan 594 . . . . . 6 ((𝜑𝜓) → (𝜓𝜒))
43biimpd 232 . . . . 5 ((𝜑𝜓) → (𝜓𝜒))
54expcom 417 . . . 4 (𝜓 → (𝜑 → (𝜓𝜒)))
65pm2.43b 55 . . 3 (𝜑 → (𝜓𝜒))
76rexlimdvw 3282 . 2 (𝜑 → (∃𝑥𝐵 𝜓𝜒))
8 rspcdv.1 . . 3 (𝜑𝐴𝐵)
98, 2rspcedv 3602 . 2 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
107, 9impbid 215 1 (𝜑 → (∃𝑥𝐵 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∃wrex 3134 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896  df-ral 3138  df-rex 3139 This theorem is referenced by:  fusgr2wsp2nb  28122
 Copyright terms: Public domain W3C validator