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Theorem ectocld 8351
Description: Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ectocl.1 𝑆 = (𝐵 / 𝑅)
ectocl.2 ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))
ectocld.3 ((𝜒𝑥𝐵) → 𝜑)
Assertion
Ref Expression
ectocld ((𝜒𝐴𝑆) → 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)

Proof of Theorem ectocld
StepHypRef Expression
1 ectocld.3 . . . 4 ((𝜒𝑥𝐵) → 𝜑)
2 ectocl.2 . . . . 5 ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))
32eqcoms 2809 . . . 4 (𝐴 = [𝑥]𝑅 → (𝜑𝜓))
41, 3syl5ibcom 248 . . 3 ((𝜒𝑥𝐵) → (𝐴 = [𝑥]𝑅𝜓))
54rexlimdva 3246 . 2 (𝜒 → (∃𝑥𝐵 𝐴 = [𝑥]𝑅𝜓))
6 elqsi 8337 . . 3 (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
7 ectocl.1 . . 3 𝑆 = (𝐵 / 𝑅)
86, 7eleq2s 2911 . 2 (𝐴𝑆 → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
95, 8impel 509 1 ((𝜒𝐴𝑆) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wrex 3110  [cec 8274   / cqs 8275
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
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-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-qs 8282
This theorem is referenced by:  ectocl  8352  elqsn0  8353  qsdisj  8361  qsel  8363  eqgen  18328  orbsta  18438  sylow1lem3  18720  sylow2alem2  18738  sylow2a  18739  sylow2blem2  18741  frgpup1  18896  frgpup3lem  18898  quscrng  20009  pi1xfr  23663  pi1coghm  23669  vitalilem3  24217  qsdisjALTV  36003  eqvrelqsel  36004
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