Theorem mpteq2da 5667

Description: Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq2da.1	⊢ Ⅎxφ
mpteq2da.2	⊢ ((φ ∧ x ∈ A) → B = C)

Assertion

Ref	Expression
mpteq2da	⊢ (φ → (x ∈ A ↦ B) = (x ∈ A ↦ C))

Proof of Theorem mpteq2da

Step	Hyp	Ref	Expression
1		eqid 2353	. . 3 ⊢ A = A
2	1	ax-gen 1546	. 2 ⊢ ∀x A = A
3		mpteq2da.1	. . 3 ⊢ Ⅎxφ
4		mpteq2da.2	. . . 4 ⊢ ((φ ∧ x ∈ A) → B = C)
5	4	ex 423	. . 3 ⊢ (φ → (x ∈ A → B = C))
6	3, 5	ralrimi 2696	. 2 ⊢ (φ → ∀x ∈ A B = C)
7		mpteq12f 5656	. 2 ⊢ ((∀x A = A ∧ ∀x ∈ A B = C) → (x ∈ A ↦ B) = (x ∈ A ↦ C))
8	2, 6, 7	sylancr 644	1 ⊢ (φ → (x ∈ A ↦ B) = (x ∈ A ↦ C))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ↦ cmpt 5652

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-opab 4624  df-mpt 5653

This theorem is used by:  mpteq2dva  5668