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Theorem bj-cleq 32588
Description: Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)
Assertion
Ref Expression
bj-cleq (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem bj-cleq
StepHypRef Expression
1 imaeq1 5424 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 eleq2 2693 . . . 4 ((𝐴𝐶) = (𝐵𝐶) → ({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
32alrimiv 1857 . . 3 ((𝐴𝐶) = (𝐵𝐶) → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
41, 3syl 17 . 2 (𝐴 = 𝐵 → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
5 abbi 2740 . 2 (∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)) ↔ {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
64, 5sylib 208 1 (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  wcel 1992  {cab 2612  {csn 4153  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by: (None)
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