Theorem fneq12d 5323

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 5321	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
3		fneq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	fneq2d 5322	. 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 1364   Fn wfn 5226

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-fun 5233  df-fn 5234

This theorem is referenced by:  fneq12  5324  tfrlemi1  6351