Theorem feq123d 5338

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 5337	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
4		feq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
5		feq3 5332	. . 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 187	1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  ⟶wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  feq123  5339  feq23d  5343