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Theorem psseq1 3008
 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 2943 . . 3 (A = B → (A𝐶B𝐶))
2 neeq1 2197 . . 3 (A = B → (A𝐶B𝐶))
31, 2anbi12d 445 . 2 (A = B → ((A𝐶 A𝐶) ↔ (B𝐶 B𝐶)))
4 df-pss 2910 . 2 (A𝐶 ↔ (A𝐶 A𝐶))
5 df-pss 2910 . 2 (B𝐶 ↔ (B𝐶 B𝐶))
63, 4, 53bitr4g 212 1 (A = B → (A𝐶B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ≠ wne 2186   ⊆ wss 2894   ⊊ wpss 2895 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 532  ax-in2 533  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ne 2188  df-in 2901  df-ss 2908  df-pss 2910 This theorem is referenced by:  psseq1i  3010  psseq1d  3013  psstr  3026  psssstr  3028
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