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Theorem 3sstr4d 3314
 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 (φA B)
3sstr4d.2 (φC = A)
3sstr4d.3 (φD = B)
Assertion
Ref Expression
3sstr4d (φC D)

Proof of Theorem 3sstr4d
StepHypRef Expression
1 3sstr4d.1 . 2 (φA B)
2 3sstr4d.2 . . 3 (φC = A)
3 3sstr4d.3 . . 3 (φD = B)
42, 3sseq12d 3300 . 2 (φ → (C DA B))
51, 4mpbird 223 1 (φC D)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  mapsspm  6021  sbthlem3  6205
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