Theorem ineqan12d 4179

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 4172	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	syl2an 596	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-in 3920

This theorem is referenced by:  funprg  6560  funtpg  6561  funcnvpr  6568  funcnvqp  6570  fvun1  6937  fndmin  7000  ofrfvalg  7630  offval  7631  offval3  7920  fpar  8053  offsplitfpar  8056  wfrlem4OLD  8263  fisn  9372  ixxin  13291  vdwmc  16861  fvcosymgeq  19225  cssincl  21129  inmbl  24943  iundisj2  24950  itg1addlem3  25099  fh1  30623  iundisj2f  31575  iundisj2fi  31768  satffunlem1lem1  34083  satffunlem2lem1  34085  disjeccnvep  36817  disjecxrn  36924  br1cosscnvxrn  37009