Theorem feq1dd 6642

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 6641	. 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 6485

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 2705

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 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6491  df-fn 6492  df-f 6493

This theorem is referenced by:  elrgspnlem4  33255  esplympl  33653  esplymhp  33654  esplyfv  33656  esplyfval3  33658  cncficcgt0  46048  itgsubsticclem  46135  itgsbtaddcnst  46142  fourierdlem103  46369  fourierdlem104  46370  fourierdlem113  46379  ismeannd  46627  hoidmv1le  46754  oppfdiag1  49575