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Theorem ectocld 6259
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 elqsi 6245 . . . 4 (𝐴 ∈ (𝐵 / 𝑅) → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
2 ectocl.1 . . . 4 𝑆 = (𝐵 / 𝑅)
31, 2eleq2s 2177 . . 3 (𝐴𝑆 → ∃𝑥𝐵 𝐴 = [𝑥]𝑅)
4 ectocld.3 . . . . 5 ((𝜒𝑥𝐵) → 𝜑)
5 ectocl.2 . . . . . 6 ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))
65eqcoms 2086 . . . . 5 (𝐴 = [𝑥]𝑅 → (𝜑𝜓))
74, 6syl5ibcom 153 . . . 4 ((𝜒𝑥𝐵) → (𝐴 = [𝑥]𝑅𝜓))
87rexlimdva 2482 . . 3 (𝜒 → (∃𝑥𝐵 𝐴 = [𝑥]𝑅𝜓))
93, 8syl5 32 . 2 (𝜒 → (𝐴𝑆𝜓))
109imp 122 1 ((𝜒𝐴𝑆) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wrex 2354  [cec 6191   / cqs 6192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-qs 6199
This theorem is referenced by:  ectocl  6260  elqsn0m  6261  qsel  6270
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