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Theorem ectocl 7702
 Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ectocl.1 𝑆 = (𝐵 / 𝑅)
ectocl.2 ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))
ectocl.3 (𝑥𝐵𝜑)
Assertion
Ref Expression
ectocl (𝐴𝑆𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)

Proof of Theorem ectocl
StepHypRef Expression
1 tru 1479 . 2
2 ectocl.1 . . 3 𝑆 = (𝐵 / 𝑅)
3 ectocl.2 . . 3 ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))
4 ectocl.3 . . . 4 (𝑥𝐵𝜑)
54adantl 481 . . 3 ((⊤ ∧ 𝑥𝐵) → 𝜑)
62, 3, 5ectocld 7701 . 2 ((⊤ ∧ 𝐴𝑆) → 𝜓)
71, 6mpan 702 1 (𝐴𝑆𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  [cec 7627   / cqs 7628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-qs 7635 This theorem is referenced by:  vitalilem2  23184
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