Theorem fneq1d 4932

Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
fneq1d.1	⊢ (φ → 𝐹 = 𝐺)

Assertion

Ref	Expression
fneq1d	⊢ (φ → (𝐹 Fn A ↔ 𝐺 Fn A))

Proof of Theorem fneq1d

Step	Hyp	Ref	Expression
1		fneq1d.1	. 2 ⊢ (φ → 𝐹 = 𝐺)
2		fneq1 4930	. 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn A ↔ 𝐺 Fn A))
3	1, 2	syl 14	1 ⊢ (φ → (𝐹 Fn A ↔ 𝐺 Fn A))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848

This theorem is referenced by:  fneq12d  4934  f1o00  5104  f1ompt  5263  fmpt2d  5270  f1ocnvd  5644  offval2  5668  ofrfval2  5669  caofinvl  5675