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

Theorem sbhypf 3003
 Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3288. (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1
sbhypf.2
Assertion
Ref Expression
sbhypf
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)   ()

Proof of Theorem sbhypf
StepHypRef Expression
1 vex 2961 . . 3
2 eqeq1 2444 . . 3
31, 2ceqsexv 2993 . 2
4 nfs1v 2184 . . . 4
5 sbhypf.1 . . . 4
64, 5nfbi 1857 . . 3
7 sbequ12 1945 . . . . 5
87bicomd 194 . . . 4
9 sbhypf.2 . . . 4
108, 9sylan9bb 682 . . 3
116, 10exlimi 1822 . 2
123, 11sylbir 206 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551  wnf 1554   wceq 1653  wsb 1659 This theorem is referenced by:  mob2  3116  tfisi  4840  ralxpf  5021  ac6sf  8371  nn0ind-raph  10372  cbvmptf  24070  ac6gf  26436  fdc1  26452 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
 Copyright terms: Public domain W3C validator