Theorem fneq12d 5422

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 5420	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
3		fneq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	fneq2d 5421	. 2 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵))
5	2, 4	bitrd 188	1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   Fn wfn 5321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-fun 5328  df-fn 5329

This theorem is referenced by:  fneq12  5423  tfrlemi1  6498