Theorem elimf 3623

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 2381, when a special case G:A-->B is provable, in order to convert F:A-->B from a hypothesis to an antecedent.

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 3617	. 2 |- (F = if(F:A-->B, F, G) -> (F:A-->B <-> if(F:A-->B, F, G):A-->B))
2		feq1 3617	. 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 2388	1 |- if(F:A-->B, F, G):A-->B

Syntax hints:  ifcif 2359  -->wf 3175

This theorem is referenced by:  ruclem39 7527  bcth 8015  ubthi 8528  hosubclt 9690  hoaddcomt 9691  hoaddasst 9699  hocsubdirt 9702  hoaddid1t 9708  hodidt 9709  ho0subt 9714  honegsubt 9716  hoddit 9906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-id 2832  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-fun 3189  df-fn 3190  df-f 3191

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