Theorem ineq12d 2218

Description: Equality deduction for intersection of two classes.

Hypotheses

Ref	Expression
ineq1d.1	|- (ph -> A = B)
ineq12d.2	|- (ph -> C = D)

Assertion

Ref	Expression
ineq12d	|- (ph -> (A i^i C) = (B i^i D))

Proof of Theorem ineq12d

Step	Hyp	Ref	Expression
1		ineq1d.1	. . 3 |- (ph -> A = B)
2	1	ineq1d 2216	. 2 |- (ph -> (A i^i C) = (B i^i C))
3		ineq12d.2	. . 3 |- (ph -> C = D)
4	3	ineq2d 2217	. 2 |- (ph -> (B i^i C) = (B i^i D))
5	2, 4	eqtrd 1507	1 |- (ph -> (A i^i C) = (B i^i D))

Syntax hints:   -> wi 3   = wceq 956   i^i cin 2046

This theorem is referenced by:  oev2 4162  mapdom2lem 4493  blin 7852  isps 8645  chocint 9418  cmbr3t 9551  pjoml3t 9555  fh1t 9561  erdisj2 10442

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

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