Theorem ineq12i 2215

Description: Equality inference for intersection of two classes. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
ineq1i.1	|- A = B
ineq12i.2	|- C = D

Assertion

Ref	Expression
ineq12i	|- (A i^i C) = (B i^i D)

Proof of Theorem ineq12i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 |- A = B
2		ineq12i.2	. 2 |- C = D
3		ineq12 2212	. 2 |- ((A = B /\ C = D) -> (A i^i C) = (B i^i D))
4	1, 2, 3	mp2an 697	1 |- (A i^i C) = (B i^i D)

Syntax hints:   = wceq 956   i^i cin 2046

This theorem is referenced by:  undir 2254  difundi 2257  difindir 2260  inrab 2271  inrab2 2272  orduniss2 3090  rnin 3458  fodomr 4483  h2hcau 8849  lejdir 9462  cmbr3 9543  nonbool 9596  5oa 9606  3oalem5 9611  mdexch 10262

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051

Copyright terms: Public domain