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Theorem ectocld 8815
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 2734 . . . 4 (𝐴 = [𝑥]𝑅 → (𝜑𝜓))
41, 3syl5ibcom 244 . . 3 ((𝜒𝑥𝐵) → (𝐴 = [𝑥]𝑅𝜓))
54rexlimdva 3145 . 2 (𝜒 → (∃𝑥𝐵 𝐴 = [𝑥]𝑅𝜓))
6 elqsi 8801 . . 3 (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
7 ectocl.1 . . 3 𝑆 = (𝐵 / 𝑅)
86, 7eleq2s 2844 . 2 (𝐴𝑆 → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
95, 8impel 504 1 ((𝜒𝐴𝑆) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  [cec 8734   / cqs 8735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-qs 8742
This theorem is referenced by:  ectocl  8816  elqsn0  8817  qsdisj  8825  qsel  8827  eqgen  19177  orbsta  19309  sylow1lem3  19600  sylow2alem2  19618  sylow2a  19619  sylow2blem2  19621  frgpup1  19775  frgpup3lem  19777  quscrng  21274  pi1xfr  25076  pi1coghm  25082  vitalilem3  25633  qsdisjALTV  38315  eqvrelqsel  38316
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