Theorem ineqan12d 4169

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 4162	. 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 395   = wceq 1541   ∩ cin 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-in 3904

This theorem is referenced by:  funprg  6535  funtpg  6536  funcnvpr  6543  funcnvqp  6545  fvun1  6913  fndmin  6978  ofrfvalg  7618  offval  7619  offval3  7914  fpar  8046  offsplitfpar  8049  fisn  9311  ixxin  13262  vdwmc  16890  fvcosymgeq  19341  cssincl  21625  inmbl  25470  iundisj2  25477  itg1addlem3  25626  fh1  31598  iundisj2f  32570  of0r  32660  iundisj2fi  32779  satffunlem1lem1  35446  satffunlem2lem1  35448  disjeccnvep  38332  disjecxrn  38446  br1cosscnvxrn  38586