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

Theorem cgsexg 3516
Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
Hypotheses
Ref Expression
cgsexg.1 (𝑥 = 𝐴𝜒)
cgsexg.2 (𝜒 → (𝜑𝜓))
Assertion
Ref Expression
cgsexg (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)   𝑉(𝑥)

Proof of Theorem cgsexg
StepHypRef Expression
1 cgsexg.2 . . . 4 (𝜒 → (𝜑𝜓))
21biimpa 479 . . 3 ((𝜒𝜑) → 𝜓)
32exlimiv 1931 . 2 (∃𝑥(𝜒𝜑) → 𝜓)
4 elisset 3484 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 cgsexg.1 . . . . 5 (𝑥 = 𝐴𝜒)
65eximi 1835 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒)
74, 6syl 17 . . 3 (𝐴𝑉 → ∃𝑥𝜒)
81biimprcd 252 . . . . 5 (𝜓 → (𝜒𝜑))
98ancld 553 . . . 4 (𝜓 → (𝜒 → (𝜒𝜑)))
109eximdv 1918 . . 3 (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒𝜑)))
117, 10syl5com 31 . 2 (𝐴𝑉 → (𝜓 → ∃𝑥(𝜒𝜑)))
123, 11impbid2 228 1 (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wex 1780  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2813  df-clel 2891
This theorem is referenced by:  ceqsexgv  3626
  Copyright terms: Public domain W3C validator