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Theorem bj-cleq 33521
 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 5715 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 eleq2 2848 . . . 4 ((𝐴𝐶) = (𝐵𝐶) → ({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
32alrimiv 1970 . . 3 ((𝐴𝐶) = (𝐵𝐶) → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
41, 3syl 17 . 2 (𝐴 = 𝐵 → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
5 abbi 2902 . 2 (∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)) ↔ {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
64, 5sylib 210 1 (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599   = wceq 1601   ∈ wcel 2107  {cab 2763  {csn 4398   “ cima 5358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368 This theorem is referenced by: (None)
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