ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  psseq1 Structured version   GIF version

Theorem psseq1 3025
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 2960 . . 3 (A = B → (A𝐶B𝐶))
2 neeq1 2213 . . 3 (A = B → (A𝐶B𝐶))
31, 2anbi12d 442 . 2 (A = B → ((A𝐶 A𝐶) ↔ (B𝐶 B𝐶)))
4 df-pss 2927 . 2 (A𝐶 ↔ (A𝐶 A𝐶))
5 df-pss 2927 . 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 1242  wne 2201  wss 2911  wpss 2912
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ne 2203  df-in 2918  df-ss 2925  df-pss 2927
This theorem is referenced by:  psseq1i  3027  psseq1d  3030  psstr  3043  psssstr  3045
  Copyright terms: Public domain W3C validator