Theorem feq3d 5502

Description: Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)

Hypothesis

Ref	Expression
feq2d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
feq3d	|- ( ph -> ( F : X --> A <-> F : X --> B ) )

Proof of Theorem feq3d

Step	Hyp	Ref	Expression
1		feq2d.1	. 2 |- ( ph -> A = B )
2		feq3 5498	. 2 |- ( A = B -> ( F : X --> A <-> F : X --> B ) )
3	1, 2	syl 14	1 |- ( ph -> ( F : X --> A <-> F : X --> B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   -->wf 5353

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227  df-f 5361

This theorem is referenced by:  gsumress  13658  resmhm2b  13744  isghm  13996  gfsumval  14102  uptx  15265  txcn  15266  dvply2g  15757  lgseisenlem3  16071  lgseisenlem4  16072  uhgr0vb  16205  uhgrun  16207  upgrun  16247  umgrun  16249  wksfval  16443  wlkres  16500