ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  psseq2 Structured version   GIF version

Theorem psseq2 3009
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 2944 . . 3 (A = B → (𝐶A𝐶B))
2 neeq2 2198 . . 3 (A = B → (𝐶A𝐶B))
31, 2anbi12d 445 . 2 (A = B → ((𝐶A 𝐶A) ↔ (𝐶B 𝐶B)))
4 df-pss 2910 . 2 (𝐶A ↔ (𝐶A 𝐶A))
5 df-pss 2910 . 2 (𝐶B ↔ (𝐶B 𝐶B))
63, 4, 53bitr4g 212 1 (A = B → (𝐶A𝐶B))
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:  psseq2i  3011  psseq2d  3014
  Copyright terms: Public domain W3C validator