Theorem elimf 6583

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4514, 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 6565	. 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵))
2		feq1 6565	. 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 4521	1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵

Syntax hints:  ifcif 4456  ⟶wf 6414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-f 6422

This theorem is referenced by:  hosubcl  30036  hoaddcom  30037  hoaddass  30045  hocsubdir  30048  hoaddid1  30054  hodid  30055  ho0sub  30060  honegsub  30062  hoddi  30253