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Theorem 3sstr4g 2980
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4g.1 (φAB)
3sstr4g.2 𝐶 = A
3sstr4g.3 𝐷 = B
Assertion
Ref Expression
3sstr4g (φ𝐶𝐷)

Proof of Theorem 3sstr4g
StepHypRef Expression
1 3sstr4g.1 . 2 (φAB)
2 3sstr4g.2 . . 3 𝐶 = A
3 3sstr4g.3 . . 3 𝐷 = B
42, 3sseq12i 2965 . 2 (𝐶𝐷AB)
51, 4sylibr 137 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:  rabss2  3017  unss2  3108  sslin  3157  ssopab2  4003  xpss12  4388  coss1  4434  coss2  4435  cnvss  4451  rnss  4507  ssres  4580  ssres2  4581  imass1  4643  imass2  4644  imadif  4922  imain  4924  ssoprab2  5503  suppssfv  5650  suppssov1  5651  tposss  5802
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