Theorem elimf 6255

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4335, 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 6237	. 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵))
2		feq1 6237	. 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 4342	1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵

Syntax hints:  ifcif 4279  ⟶wf 6097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-fun 6103  df-fn 6104  df-f 6105

This theorem is referenced by:  hosubcl  28960  hoaddcom  28961  hoaddass  28969  hocsubdir  28972  hoaddid1  28978  hodid  28979  ho0sub  28984  honegsub  28986  hoddi  29177