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Theorem sbcco2 2780
Description: A composition law for class substitution. Importantly, may occur free in the class expression substituted for . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcco2.1
Assertion
Ref Expression
sbcco2  [.  ]. [.  ].  [.  ].
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem sbcco2
StepHypRef Expression
1 sbsbc 2762 . 2  [.  ].  [.  ]. [.  ].
2 nfv 1418 . . 3  F/
[.  ].
3 sbcco2.1 . . . . 5
43equcoms 1591 . . . 4
5 dfsbcq 2760 . . . . 5  [.  ]. 
[.  ].
65bicomd 129 . . . 4  [.  ]. 
[.  ].
74, 6syl 14 . . 3  [.  ]. 
[.  ].
82, 7sbie 1671 . 2  [.  ].  [.  ].
91, 8bitr3i 175 1  [.  ]. [.  ].  [.  ].
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242  wsb 1642   [.wsbc 2758
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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  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-sbc 2759
This theorem is referenced by: (None)
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