Theorem feq123d 6651

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 6650	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
4		feq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
5	4	feq3d 6647	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))
6	3, 5	bitrd 279	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  feq123  6652  feq23d  6657  fprg  7102  csbwrdg  14497  funcestrcsetclem8  18104  funcsetcestrclem8  18119  funcsetcestrclem9  18120  evlfcl  18179  yonedalem3a  18231  yonedalem4c  18234  yonedalem3b  18236  yonedainv  18238  iscau  25253  isuhgr  29143  uhgreq12g  29148  isuhgrop  29153  uhgrun  29157  isupgr  29167  upgrop  29177  isumgr  29178  upgrun  29201  umgrun  29203  lfuhgr1v0e  29337  wlkp1  29763  sseqf  34552  ismfs  35747  isrngo  38232  gneispace2  44577  isubgruhgr  48356  funcringcsetcALTV2lem8  48785  funcringcsetclem8ALTV  48808