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Theorem 3sstr4d 2982
 Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4d.1 (φAB)
3sstr4d.2 (φ𝐶 = A)
3sstr4d.3 (φ𝐷 = B)
Assertion
Ref Expression
3sstr4d (φ𝐶𝐷)

Proof of Theorem 3sstr4d
StepHypRef Expression
1 3sstr4d.1 . 2 (φAB)
2 3sstr4d.2 . . 3 (φ𝐶 = A)
3 3sstr4d.3 . . 3 (φ𝐷 = B)
42, 3sseq12d 2968 . 2 (φ → (𝐶𝐷AB))
51, 4mpbird 156 1 (φ𝐶𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ⊆ wss 2911 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-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-in 2918  df-ss 2925 This theorem is referenced by:  rdgss  5910  sucinc2  5965  oawordi  5988  fzoss1  8777  fzoss2  8778
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