Theorem elimf 6701

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4541, 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 6680	. 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵))
2		feq1 6680	. 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 4548	1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵

This proof depends on syntax axioms:  ifcif 4482  ⟶wf 6529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6535  df-fn 6536  df-f 6537

This theorem is used by:  hosubcl  32254  hoaddcom  32255  hoaddass  32263  hocsubdir  32266  hoaddrid  32272  hodid  32273  ho0sub  32278  honegsub  32280  hoddi  32471