Theorem feq1dd 6669

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 6668	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
4	1, 3	mpbid 234	1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)

Syntax hints:   → wi 4   = wceq 1559  ⟶wf 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6518  df-fn 6519  df-f 6520

This theorem is referenced by:  elrgspnlem4  33387  0mplrim  33772  esplympl  33825  esplymhp  33826  esplyfv  33828  esplyfval3  33830  cncficcgt0  46423  itgsubsticclem  46510  itgsbtaddcnst  46517  fourierdlem103  46744  fourierdlem104  46745  fourierdlem113  46754  ismeannd  47002  hoidmv1le  47129  sssmf  47273  oppfdiag1  49996