Theorem feq123d 6720

Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
feq12d.1	⊢ (𝜑 → 𝐹 = 𝐺)
feq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)
feq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
feq123d	⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))

Proof of Theorem feq123d

Step	Hyp	Ref	Expression
1		feq12d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		feq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	1, 2	feq12d 6719	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
4		feq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
5	4	feq3d 6716	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))
6	3, 5	bitrd 282	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ⟶wf 6554

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6560  df-fn 6561  df-f 6562

This theorem is used by:  feq123  6721  feq23d  6726  fprg  7183  csbwrdg  14716  funcestrcsetclem8  18350  funcsetcestrclem8  18365  funcsetcestrclem9  18366  evlfcl  18425  yonedalem3a  18477  yonedalem4c  18480  yonedalem3b  18482  yonedainv  18484  iscau  25627  isuhgr  29668  uhgreq12g  29673  isuhgrop  29678  uhgrun  29682  isupgr  29692  upgrop  29702  isumgr  29703  upgrun  29726  umgrun  29728  lfuhgr1v0e  29865  wlkp1  30290  sseqf  35094  ismfs  36481  isrngo  39023  gneispace2  45324  isubgruhgr  49171  funcringcsetcALTV2lem8  49599  funcringcsetclem8ALTV  49622  ixpovf  50402