Theorem ineqan12d 4122

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 4115	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
4	1, 2, 3	syl2an 608	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-in 4094

This theorem is used by:  funprg  6605  funtpg  6606  funcnvpr  6613  funcnvqp  6615  fvun1  6987  fndmin  7055  ofrfvalg  7711  offval  7712  offval3  8004  fpar  8140  offsplitfpar  8143  fisn  9432  ixxin  13506  vdwmc  17171  fvcosymgeq  19658  cssincl  22009  inmbl  25878  iundisj2  25885  itg1addlem3  26034  fh1  32235  iundisj2f  33241  of0r  33330  iundisj2fi  33445  satffunlem1lem1  36291  satffunlem2lem1  36293  disjeccnvep  39365  disjecxrn  39487  br1cosscnvxrn  39639