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Theorem ectocl 6578
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 1352 . 2
2 ectocl.1 . . 3 𝑆 = (𝐵 / 𝑅)
3 ectocl.2 . . 3 ([𝑥]𝑅 = 𝐴 → (𝜑𝜓))
4 ectocl.3 . . . 4 (𝑥𝐵𝜑)
54adantl 275 . . 3 ((⊤ ∧ 𝑥𝐵) → 𝜑)
62, 3, 5ectocld 6577 . 2 ((⊤ ∧ 𝐴𝑆) → 𝜓)
71, 6mpan 422 1 (𝐴𝑆𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  wtru 1349  wcel 2141  [cec 6509   / cqs 6510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-qs 6517
This theorem is referenced by: (None)
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