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Theorem ectocl 6109
 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 (x Bφ)
Assertion
Ref Expression
ectocl (A 𝑆ψ)
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 1246 . 2
2 ectocl.1 . . 3 𝑆 = (B / 𝑅)
3 ectocl.2 . . 3 ([x]𝑅 = A → (φψ))
4 ectocl.3 . . . 4 (x Bφ)
54adantl 262 . . 3 (( ⊤ x B) → φ)
62, 3, 5ectocld 6108 . 2 (( ⊤ A 𝑆) → ψ)
71, 6mpan 400 1 (A 𝑆ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ⊤ wtru 1243   ∈ wcel 1390  [cec 6040   / cqs 6041 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-qs 6048 This theorem is referenced by: (None)
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