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Theorem bj-cleq 36334
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 6045 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 eleq2 2814 . . 3 ((𝐴𝐶) = (𝐵𝐶) → ({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
32alrimiv 1922 . 2 ((𝐴𝐶) = (𝐵𝐶) → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
4 abbi 2792 . 2 (∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)) → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
51, 3, 43syl 18 1 (𝐴 = 𝐵 → {𝑥 ∣ {𝑥} ∈ (𝐴𝐶)} = {𝑥 ∣ {𝑥} ∈ (𝐵𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wcel 2098  {cab 2701  {csn 4621  cima 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680
This theorem is referenced by: (None)
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