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Theorem ectocld 5991
 Description: Implicit substitution of class for equivalence class. (Contributed by set.mm contributors, 9-Jul-2014.)
Hypotheses
Ref Expression
ectocl.1 S = (B / R)
ectocl.2 ([x]R = A → (φψ))
ectocld.3 ((χ x B) → φ)
Assertion
Ref Expression
ectocld ((χ A S) → ψ)
Distinct variable groups:   x,A   x,B   x,R   ψ,x   χ,x
Allowed substitution hints:   φ(x)   S(x)

Proof of Theorem ectocld
StepHypRef Expression
1 elqsi 5978 . . . 4 (A (B / R) → x B A = [x]R)
2 ectocl.1 . . . 4 S = (B / R)
31, 2eleq2s 2445 . . 3 (A Sx B A = [x]R)
4 ectocld.3 . . . . 5 ((χ x B) → φ)
5 ectocl.2 . . . . . 6 ([x]R = A → (φψ))
65eqcoms 2356 . . . . 5 (A = [x]R → (φψ))
74, 6syl5ibcom 211 . . . 4 ((χ x B) → (A = [x]Rψ))
87rexlimdva 2738 . . 3 (χ → (x B A = [x]Rψ))
93, 8syl5 28 . 2 (χ → (A Sψ))
109imp 418 1 ((χ A S) → ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  [cec 5945   / cqs 5946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  ectocl  5992  elqsn0  5993  qsdisj  5995
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