Theorem ineq12d 1646

Description: Equality deduction for intersection of two classes.

Hypotheses

Ref	Expression
ineq1d.1	⊢ (φ → A = B)
ineq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
ineq12d	⊢ (φ → (A ∩ C) = (B ∩ D))

Proof of Theorem ineq12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. . 3 ⊢ (φ → A = B)
2	1	ineq1d 1644	. 2 ⊢ (φ → (A ∩ C) = (B ∩ C))
3		ineq12d.2	. . 3 ⊢ (φ → C = D)
4	3	ineq2d 1645	. 2 ⊢ (φ → (B ∩ C) = (B ∩ D))
5	2, 4	eqtrd 1128	1 ⊢ (φ → (A ∩ C) = (B ∩ D))

Syntax hints:   → wi 2   = wceq 1091   ∩ cin 1486

This theorem is referenced by:  mapdom2lem 3388  pjoml3t 5517

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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