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Theorem ectocl 8776
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 1537 . 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 8775 . 2 ((⊤ ∧ 𝐴𝑆) → 𝜓)
71, 6mpan 687 1 (𝐴𝑆𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wtru 1534  wcel 2098  [cec 8698   / cqs 8699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-qs 8706
This theorem is referenced by:  vitalilem2  25482
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