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Theorem ectocld 7970
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 2779 . . . 4 (𝐴 = [𝑥]𝑅 → (𝜑𝜓))
41, 3syl5ibcom 235 . . 3 ((𝜒𝑥𝐵) → (𝐴 = [𝑥]𝑅𝜓))
54rexlimdva 3179 . 2 (𝜒 → (∃𝑥𝐵 𝐴 = [𝑥]𝑅𝜓))
6 elqsi 7956 . . 3 (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
7 ectocl.1 . . 3 𝑆 = (𝐵 / 𝑅)
86, 7eleq2s 2868 . 2 (𝐴𝑆 → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
95, 8impel 495 1 ((𝜒𝐴𝑆) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  [cec 7898   / cqs 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-qs 7906
This theorem is referenced by:  ectocl  7971  elqsn0  7972  qsdisj  7980  qsel  7982  eqgen  17855  orbsta  17953  sylow1lem3  18222  sylow2alem2  18240  sylow2a  18241  sylow2blem2  18243  frgpup1  18395  frgpup3lem  18397  quscrng  19455  pi1xfr  23074  pi1coghm  23080  vitalilem3  23598
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