Theorem feq1dd 6701

Description: Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
feq1dd.eq	⊢ (𝜑 → 𝐹 = 𝐺)
feq1dd.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
feq1dd	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)

Proof of Theorem feq1dd

Step	Hyp	Ref	Expression
1		feq1dd.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		feq1dd.eq	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
3	2	feq1d 6700	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)

Syntax hints:   → wi 4   = wceq 1539  ⟶wf 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-fun 6543  df-fn 6544  df-f 6545

This theorem is referenced by:  elrgspnlem4  33188  cncficcgt0  45860  itgsubsticclem  45947  itgsbtaddcnst  45954  fourierdlem103  46181  fourierdlem104  46182  fourierdlem113  46191  ismeannd  46439  hoidmv1le  46566