Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ectocl GIF version

Theorem ectocl 6496
 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 1335 . 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 6495 . 2 ((⊤ ∧ 𝐴𝑆) → 𝜓)
71, 6mpan 420 1 (𝐴𝑆𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ⊤wtru 1332   ∈ wcel 1480  [cec 6427   / cqs 6428 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-qs 6435 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator