Theorem elimf 6710

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4564, 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 6691	. 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵))
2		feq1 6691	. 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 4571	1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵

Syntax hints:  ifcif 4505  ⟶wf 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  hosubcl  31759  hoaddcom  31760  hoaddass  31768  hocsubdir  31771  hoaddrid  31777  hodid  31778  ho0sub  31783  honegsub  31785  hoddi  31976