Theorem mpt2eq3ia 5670

Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpt2eq3ia.1	⊢ ((x ∈ A ∧ y ∈ B) → C = D)

Assertion

Ref	Expression
mpt2eq3ia	⊢ (x ∈ A, y ∈ B ↦ C) = (x ∈ A, y ∈ B ↦ D)

Proof of Theorem mpt2eq3ia

Step	Hyp	Ref	Expression
1		mpt2eq3ia.1	. . . 4 ⊢ ((x ∈ A ∧ y ∈ B) → C = D)
2	1	3adant1 973	. . 3 ⊢ (( ⊤ ∧ x ∈ A ∧ y ∈ B) → C = D)
3	2	mpt2eq3dva 5669	. 2 ⊢ ( ⊤ → (x ∈ A, y ∈ B ↦ C) = (x ∈ A, y ∈ B ↦ D))
4	3	trud 1323	1 ⊢ (x ∈ A, y ∈ B ↦ C) = (x ∈ A, y ∈ B ↦ D)

Syntax hints:   → wi 4   ∧ wa 358   ⊤ wtru 1316   = wceq 1642   ∈ wcel 1710   ↦ cmpt2 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-oprab 5528  df-mpt2 5654

This theorem is referenced by: (None)