Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbccomieg Structured version   Unicode version

Theorem sbccomieg 26849
 Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
sbccomieg.1
Assertion
Ref Expression
sbccomieg
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   (,)   ()

Proof of Theorem sbccomieg
StepHypRef Expression
1 sbcex 3170 . 2
2 spesbc 3242 . . 3
3 sbcex 3170 . . . 4
43exlimiv 1644 . . 3
52, 4syl 16 . 2
6 nfcv 2572 . . . 4
7 nfsbc1v 3180 . . . 4
86, 7nfsbc 3182 . . 3
9 sbccomieg.1 . . . . 5
10 dfsbcq 3163 . . . . 5
119, 10syl 16 . . . 4
12 sbceq1a 3171 . . . . 5
1312sbcbidv 3215 . . . 4
1411, 13bitrd 245 . . 3
158, 14sbciegf 3192 . 2
161, 5, 15pm5.21nii 343 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177  wex 1550   wceq 1652   wcel 1725  cvv 2956  wsbc 3161 This theorem is referenced by:  sbccomiegOLD  26853  2rexfrabdioph  26856  3rexfrabdioph  26857  4rexfrabdioph  26858  6rexfrabdioph  26859  7rexfrabdioph  26860 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-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162
 Copyright terms: Public domain W3C validator