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Theorem psseq1 3059
 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 2994 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 neeq1 2233 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
31, 2anbi12d 450 . 2 (𝐴 = 𝐵 → ((𝐴𝐶𝐴𝐶) ↔ (𝐵𝐶𝐵𝐶)))
4 df-pss 2961 . 2 (𝐴𝐶 ↔ (𝐴𝐶𝐴𝐶))
5 df-pss 2961 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵𝐶))
63, 4, 53bitr4g 216 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ≠ wne 2220   ⊆ wss 2945   ⊊ wpss 2946 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ne 2221  df-in 2952  df-ss 2959  df-pss 2961 This theorem is referenced by:  psseq1i  3061  psseq1d  3064  psstr  3077  psssstr  3079
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