Theorem fneq12d 5412

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

Hypotheses

Ref	Expression
fneq12d.1	|- ( ph -> F = G )
fneq12d.2	|- ( ph -> A = B )

Assertion

Ref	Expression
fneq12d	|- ( ph -> ( F Fn A <-> G Fn B ) )

Proof of Theorem fneq12d

Step	Hyp	Ref	Expression
1		fneq12d.1	. . 3 |- ( ph -> F = G )
2	1	fneq1d 5410	. 2 |- ( ph -> ( F Fn A <-> G Fn A ) )
3		fneq12d.2	. . 3 |- ( ph -> A = B )
4	3	fneq2d 5411	. 2 |- ( ph -> ( G Fn A <-> G Fn B ) )
5	2, 4	bitrd 188	1 |- ( ph -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   Fn wfn 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-fun 5319  df-fn 5320

This theorem is referenced by:  fneq12  5413  tfrlemi1  6476