Theorem fneq12d 5292

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 5290	. 2 |- ( ph -> ( F Fn A <-> G Fn A ) )
3		fneq12d.2	. . 3 |- ( ph -> A = B )
4	3	fneq2d 5291	. 2 |- ( ph -> ( G Fn A <-> G Fn B ) )
5	2, 4	bitrd 187	1 |- ( ph -> ( F Fn A <-> G Fn B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1349   Fn wfn 5195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991  df-opab 4052  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-fun 5202  df-fn 5203

This theorem is referenced by:  fneq12  5293  tfrlemi1  6315