Theorem ineqan12d 4168

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 4161	. 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 1570   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-in 3906

This theorem is used by:  funprg  6594  funtpg  6595  funcnvpr  6602  funcnvqp  6604  fvun1  6976  fndmin  7044  ofrfvalg  7701  offval  7702  offval3  7994  fpar  8127  offsplitfpar  8130  fisn  9419  ixxin  13493  vdwmc  17156  fvcosymgeq  19643  cssincl  21994  inmbl  25863  iundisj2  25870  itg1addlem3  26019  fh1  32220  iundisj2f  33184  of0r  33273  iundisj2fi  33389  satffunlem1lem1  36167  satffunlem2lem1  36169  disjeccnvep  39222  disjecxrn  39344  br1cosscnvxrn  39496