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Theorem ectocld 6083
Description: Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 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 6069 . . . 4 (A (B / 𝑅) → x B A = [x]𝑅)
2 ectocl.1 . . . 4 𝑆 = (B / 𝑅)
31, 2eleq2s 2114 . . 3 (A 𝑆x B A = [x]𝑅)
4 ectocld.3 . . . . 5 ((χ x B) → φ)
5 ectocl.2 . . . . . 6 ([x]𝑅 = A → (φψ))
65eqcoms 2025 . . . . 5 (A = [x]𝑅 → (φψ))
74, 6syl5ibcom 144 . . . 4 ((χ x B) → (A = [x]𝑅ψ))
87rexlimdva 2411 . . 3 (χ → (x B A = [x]𝑅ψ))
93, 8syl5 28 . 2 (χ → (A 𝑆ψ))
109imp 115 1 ((χ A 𝑆) → ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wrex 2285  [cec 6015   / cqs 6016
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-qs 6023
This theorem is referenced by:  ectocl  6084  elqsn0m  6085  qsel  6094
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