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Theorem 3sstr3d 2987
 Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
Hypotheses
Ref Expression
3sstr3d.1 (𝜑𝐴𝐵)
3sstr3d.2 (𝜑𝐴 = 𝐶)
3sstr3d.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
3sstr3d (𝜑𝐶𝐷)

Proof of Theorem 3sstr3d
StepHypRef Expression
1 3sstr3d.1 . 2 (𝜑𝐴𝐵)
2 3sstr3d.2 . . 3 (𝜑𝐴 = 𝐶)
3 3sstr3d.3 . . 3 (𝜑𝐵 = 𝐷)
42, 3sseq12d 2974 . 2 (𝜑 → (𝐴𝐵𝐶𝐷))
51, 4mpbid 135 1 (𝜑𝐶𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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