Theorem mpteq12dv 5657

Description: An equality inference for the maps to notation. (Contributed by set.mm contributors, 24-Aug-2011.) (Revised by set.mm contributors, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12dv.1	⊢ (φ → A = C)
mpteq12dv.2	⊢ (φ → B = D)

Assertion

Ref	Expression
mpteq12dv	⊢ (φ → (x ∈ A ↦ B) = (x ∈ C ↦ D))

Distinct variable group:   φ,x

Allowed substitution hints:   A(x)   B(x)   C(x)   D(x)

Proof of Theorem mpteq12dv

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. . 3 ⊢ (φ → A = C)
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x A = C)
3		mpteq12dv.2	. . 3 ⊢ (φ → B = D)
4	3	ralrimivw 2699	. 2 ⊢ (φ → ∀x ∈ A B = D)
5		mpteq12f 5656	. 2 ⊢ ((∀x A = C ∧ ∀x ∈ A B = D) → (x ∈ A ↦ B) = (x ∈ C ↦ D))
6	2, 4, 5	syl2anc 642	1 ⊢ (φ → (x ∈ A ↦ B) = (x ∈ C ↦ D))

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642  ∀wral 2615   ↦ cmpt 5652

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-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 referenced by:  mpteq12i  5666