Theorem feq3d 5420

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 5416	. 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 1373  ⟶wf 5272

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3173  df-ss 3180  df-f 5280

This theorem is referenced by:  gsumress  13271  resmhm2b  13365  isghm  13623  uptx  14790  txcn  14791  dvply2g  15282  lgseisenlem3  15593  lgseisenlem4  15594  uhgr0vb  15724  uhgrun  15726