Theorem ineq12i 2205

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 2202	. 2 |- ((A = B /\ C = D) -> (A i^i C) = (B i^i D))
4	1, 2, 3	mp2an 695	1 |- (A i^i C) = (B i^i D)

Syntax hints:   = wceq 953   i^i cin 2036

This theorem is referenced by:  undir 2244  difundi 2247  difindir 2250  inrab 2261  inrab2 2262  orduniss2 3080  rnin 3444  fodomr 4463  h2hcau 8788  lejdir 9377  cmbr3 9460  nonbool 9513  5oa 9523  3oalem5 9528  mdexch 10170

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041

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