Theorem fneq12d 5385

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 5383	. 2 |- ( ph -> ( F Fn A <-> G Fn A ) )
3		fneq12d.2	. . 3 |- ( ph -> A = B )
4	3	fneq2d 5384	. 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 1373   Fn wfn 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-fun 5292  df-fn 5293

This theorem is referenced by:  fneq12  5386  tfrlemi1  6441