Theorem feq3d 5522

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 5518	. 2 |- ( A = B -> ( F : X --> A <-> F : X --> B ) )
3	1, 2	syl 14	1 |- ( ph -> ( F : X --> A <-> F : X --> B ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5381

This theorem is used by:  gzsumress  13712  resmhm2b  13796  isghm  14046  gsumvalfi  14152  gsumressfi  14167  uptx  15375  txcn  15376  dvply2g  15867  lgseisenlem3  16191  lgseisenlem4  16192  uhgr0vb  16325  uhgrun  16327  upgrun  16367  umgrun  16369  wksfval  16563  wlkres  16620