Theorem feq1dd 6645

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 6644	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)

Syntax hints:   → wi 4   = wceq 1541  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  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:  elrgspnlem4  33327  esplympl  33725  esplymhp  33726  esplyfv  33728  esplyfval3  33730  cncficcgt0  46132  itgsubsticclem  46219  itgsbtaddcnst  46226  fourierdlem103  46453  fourierdlem104  46454  fourierdlem113  46463  ismeannd  46711  hoidmv1le  46838  oppfdiag1  49659