Theorem ineqan12d 2215

Description: Equality deduction for intersection of two classes.

Hypotheses

Ref	Expression
ineq1d.1	|- (ph -> A = B)
ineqan12d.2	|- (ps -> C = D)

Assertion

Ref	Expression
ineqan12d	|- ((ph /\ ps) -> (A i^i C) = (B i^i D))

Proof of Theorem ineqan12d

Step	Hyp	Ref	Expression
1		ineq12 2208	. 2 |- ((A = B /\ C = D) -> (A i^i C) = (B i^i D))
2		ineq1d.1	. 2 |- (ph -> A = B)
3		ineqan12d.2	. 2 |- (ps -> C = D)
4	1, 2, 3	syl2an 454	1 |- ((ph /\ ps) -> (A i^i C) = (B i^i D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   i^i cin 2042

This theorem is referenced by:  iooint 6317  fh1t 9501

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047

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