Theorem elimf 6044

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4139, when a special case G : A --> B is provable, in order to convert F : A --> B from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)

Hypothesis

Ref	Expression
elimf.1	|- G : A --> B

Assertion

Ref	Expression
elimf	|- if ( F : A --> B , F , G ) : A --> B

Proof of Theorem elimf

Step	Hyp	Ref	Expression
1		feq1 6026	. 2 |- ( F = if ( F : A --> B , F , G ) -> ( F : A --> B <-> if ( F : A --> B , F , G ) : A --> B ) )
2		feq1 6026	. 2 |- ( G = if ( F : A --> B , F , G ) -> ( G : A --> B <-> if ( F : A --> B , F , G ) : A --> B ) )
3		elimf.1	. 2 |- G : A --> B
4	1, 2, 3	elimhyp 4146	1 |- if ( F : A --> B , F , G ) : A --> B

Syntax hints:   ifcif 4086   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  hosubcl  28632  hoaddcom  28633  hoaddass  28641  hocsubdir  28644  hoaddid1  28650  hodid  28651  ho0sub  28656  honegsub  28658  hoddi  28849