Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbabel Structured version   Unicode version

Theorem sbabel 2598
 Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
sbabel.1
Assertion
Ref Expression
sbabel
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem sbabel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbex 2205 . . 3
2 sban 2139 . . . . 5
3 nfv 1629 . . . . . . . . . 10
43sbf 2117 . . . . . . . . 9
54sbrbis 2145 . . . . . . . 8
65sbalv 2206 . . . . . . 7
7 abeq2 2541 . . . . . . . 8
87sbbii 1665 . . . . . . 7
9 abeq2 2541 . . . . . . 7
106, 8, 93bitr4i 269 . . . . . 6
11 sbabel.1 . . . . . . . 8
1211nfcri 2566 . . . . . . 7
1312sbf 2117 . . . . . 6
1410, 13anbi12i 679 . . . . 5
152, 14bitri 241 . . . 4
1615exbii 1592 . . 3
171, 16bitri 241 . 2
18 df-clel 2432 . . 3
1918sbbii 1665 . 2
20 df-clel 2432 . 2
2117, 19, 203bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wal 1549  wex 1550   wceq 1652  wsb 1658   wcel 1725  cab 2422  wnfc 2559 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561
 Copyright terms: Public domain W3C validator