Theorem feq123d 5057

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 5056	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
4		feq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
5		feq3 5052	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))
7	3, 6	bitrd 186	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284  ⟶wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926

This theorem is referenced by:  feq123  5058  feq23d  5062