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Theorem psseq1 3031
Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psseq1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Proof of Theorem psseq1
StepHypRef Expression
1 sseq1 2966 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 neeq1 2218 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
31, 2anbi12d 442 . 2 (𝐴 = 𝐵 → ((𝐴𝐶𝐴𝐶) ↔ (𝐵𝐶𝐵𝐶)))
4 df-pss 2933 . 2 (𝐴𝐶 ↔ (𝐴𝐶𝐴𝐶))
5 df-pss 2933 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵𝐶))
63, 4, 53bitr4g 212 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wne 2204  wss 2917  wpss 2918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933
This theorem is referenced by:  psseq1i  3033  psseq1d  3036  psstr  3049  psssstr  3051
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