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Theorem elimf 5942
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4088, when a special case 𝐺:𝐴𝐵 is provable, in order to convert 𝐹:𝐴𝐵 from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
Hypothesis
Ref Expression
elimf.1 𝐺:𝐴𝐵
Assertion
Ref Expression
elimf if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵

Proof of Theorem elimf
StepHypRef Expression
1 feq1 5924 . 2 (𝐹 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐹:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
2 feq1 5924 . 2 (𝐺 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐺:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
3 elimf.1 . 2 𝐺:𝐴𝐵
41, 2, 3elimhyp 4095 1 if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ifcif 4035  wf 5785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-fun 5791  df-fn 5792  df-f 5793
This theorem is referenced by:  hosubcl  27809  hoaddcom  27810  hoaddass  27818  hocsubdir  27821  hoaddid1  27827  hodid  27828  ho0sub  27833  honegsub  27835  hoddi  28026
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