Theorem ineqan12d 2799

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 2791	. 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 503	1 |- ((ph /\ ps) -> (A i^i C) = (B i^i D))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   i^i cin 2592

This theorem is referenced by:  fpar 5085  iooin 7539  fh1 11194  wfrlem4 13960

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603

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