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 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-in 3906

This theorem is used by:  funprg  6588  funtpg  6589  funcnvpr  6596  funcnvqp  6598  fvun1  6970  fndmin  7038  ofrfvalg  7687  offval  7688  offval3  7980  fpar  8114  offsplitfpar  8117  fisn  9400  ixxin  13418  vdwmc  17073  fvcosymgeq  19559  cssincl  21904  inmbl  25773  iundisj2  25780  itg1addlem3  25929  fh1  32102  iundisj2f  33066  of0r  33155  iundisj2fi  33271  satffunlem1lem1  35984  satffunlem2lem1  35986  disjeccnvep  39041  disjecxrn  39163  br1cosscnvxrn  39315