Theorem ineqan12d 4151

Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)

Hypotheses

Ref	Expression
ineq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
ineqan12d.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
ineqan12d	⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Proof of Theorem ineqan12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ineqan12d.2	. 2 ⊢ (𝜓 → 𝐶 = 𝐷)
3		ineq12 4144	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	syl2an 602	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-in 3890

This theorem is referenced by:  funprg  6539  funtpg  6540  funcnvpr  6547  funcnvqp  6549  fvun1  6918  fndmin  6986  ofrfvalg  7628  offval  7629  offval3  7924  fpar  8055  offsplitfpar  8058  fisn  9330  ixxin  13306  vdwmc  16940  fvcosymgeq  19395  cssincl  21663  inmbl  25527  iundisj2  25534  itg1addlem3  25683  fh1  31707  iundisj2f  32679  of0r  32771  iundisj2fi  32889  satffunlem1lem1  35630  satffunlem2lem1  35632  disjeccnvep  38657  disjecxrn  38779  br1cosscnvxrn  38931