Theorem feq1dd 6686

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 6685	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)

Syntax hints:   → wi 4   = wceq 1567  ⟶wf 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6536  df-fn 6537  df-f 6538

This theorem is referenced by:  elrgspnlem4  33502  0mplrim  33845  esplympl  33898  esplymhp  33899  esplyfv  33901  esplyfval3  33903  cncficcgt0  46489  itgsubsticclem  46576  itgsbtaddcnst  46583  fourierdlem103  46810  fourierdlem104  46811  fourierdlem113  46820  ismeannd  47068  hoidmv1le  47195  sssmf  47339  oppfdiag1  50072