Theorem fneq12d 5351

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 5349	. 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
3		fneq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	fneq2d 5350	. 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 5254

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-fun 5261  df-fn 5262

This theorem is referenced by:  fneq12  5352  tfrlemi1  6399