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Theorem bj-cleq 35152
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 5964 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 eleq2 2827 . . 3 ((𝐴𝐶) = (𝐵𝐶) → ({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
32alrimiv 1930 . 2 ((𝐴𝐶) = (𝐵𝐶) → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
4 abbi1 2806 . 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 1537   = wceq 1539  wcel 2106  {cab 2715  {csn 4561  cima 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602
This theorem is referenced by: (None)
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