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Theorem bj-cleq 34262
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 5917 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 eleq2 2899 . . 3 ((𝐴𝐶) = (𝐵𝐶) → ({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
32alrimiv 1921 . 2 ((𝐴𝐶) = (𝐵𝐶) → ∀𝑥({𝑥} ∈ (𝐴𝐶) ↔ {𝑥} ∈ (𝐵𝐶)))
4 abbi1 2882 . 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 208  wal 1528   = wceq 1530  wcel 2107  {cab 2797  {csn 4559  cima 5551
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by: (None)
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