Theorem feq3d 5462

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

Hypothesis

Ref	Expression
feq2d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
feq3d	⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))

Proof of Theorem feq3d

Step	Hyp	Ref	Expression
1		feq2d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		feq3 5458	. 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ⟶wf 5314

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  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-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-f 5322

This theorem is referenced by:  gsumress  13436  resmhm2b  13530  isghm  13788  uptx  14956  txcn  14957  dvply2g  15448  lgseisenlem3  15759  lgseisenlem4  15760  uhgr0vb  15892  uhgrun  15894  upgrun  15932  umgrun  15934  wksfval  16043  wlkres  16098