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Theorem ectocld 6108
 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 6094 . . . 4 (A (B / 𝑅) → x B A = [x]𝑅)
2 ectocl.1 . . . 4 𝑆 = (B / 𝑅)
31, 2eleq2s 2129 . . 3 (A 𝑆x B A = [x]𝑅)
4 ectocld.3 . . . . 5 ((χ x B) → φ)
5 ectocl.2 . . . . . 6 ([x]𝑅 = A → (φψ))
65eqcoms 2040 . . . . 5 (A = [x]𝑅 → (φψ))
74, 6syl5ibcom 144 . . . 4 ((χ x B) → (A = [x]𝑅ψ))
87rexlimdva 2427 . . 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 1242   ∈ wcel 1390  ∃wrex 2301  [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-bndl 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:  ectocl  6109  elqsn0m  6110  qsel  6119
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