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Theorem psseq2 3032
 Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psseq2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem psseq2
StepHypRef Expression
1 sseq2 2967 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
2 neeq2 2219 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
31, 2anbi12d 442 . 2 (𝐴 = 𝐵 → ((𝐶𝐴𝐶𝐴) ↔ (𝐶𝐵𝐶𝐵)))
4 df-pss 2933 . 2 (𝐶𝐴 ↔ (𝐶𝐴𝐶𝐴))
5 df-pss 2933 . 2 (𝐶𝐵 ↔ (𝐶𝐵𝐶𝐵))
63, 4, 53bitr4g 212 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 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:  psseq2i  3034  psseq2d  3037
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