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Theorem ectocl 5992
 Description: Implicit substitution of class for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 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 1321 . 2
2 ectocl.1 . . 3 S = (B / R)
3 ectocl.2 . . 3 ([x]R = A → (φψ))
4 ectocl.3 . . . 4 (x Bφ)
54adantl 452 . . 3 (( ⊤ x B) → φ)
62, 3, 5ectocld 5991 . 2 (( ⊤ A S) → ψ)
71, 6mpan 651 1 (A Sψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ⊤ wtru 1316   = wceq 1642   ∈ wcel 1710  [cec 5945   / cqs 5946 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-qs 5951 This theorem is referenced by: (None)
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