Theorem elimf 6515

Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4525, 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 6497	. 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵))
2		feq1 6497	. 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 4532	1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵

Syntax hints:  ifcif 4469  ⟶wf 6353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-f 6361

This theorem is referenced by:  hosubcl  29552  hoaddcom  29553  hoaddass  29561  hocsubdir  29564  hoaddid1  29570  hodid  29571  ho0sub  29576  honegsub  29578  hoddi  29769