Theorem fneq12d 6644

Description: Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)

Hypotheses

Ref	Expression
fneq12d.1	⊢ (𝜑 → 𝐹 = 𝐺)
fneq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
fneq12d	⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵))

Proof of Theorem fneq12d

Step	Hyp	Ref	Expression
1		fneq12d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2	1	fneq1d 6642	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
3		fneq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	fneq2d 6643	. 2 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵))
5	2, 4	bitrd 279	1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   Fn wfn 6538

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-fun 6545  df-fn 6546

This theorem is referenced by:  fneq12  6645  seqfn  13983  sscres  17775  reschomf  17784  funcres  17851  psrvscafval  21729  ressprdsds  24098  rrxmfval  25155  ex-fpar  29983  sseqfn  33688  tfsconcatfn  42391  funcoressn  46051