Theorem ineq12d 2221

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 2219	. 2 |- (ph -> (A i^i C) = (B i^i C))
3		ineq12d.2	. . 3 |- (ph -> C = D)
4	3	ineq2d 2220	. 2 |- (ph -> (B i^i C) = (B i^i D))
5	2, 4	eqtrd 1510	1 |- (ph -> (A i^i C) = (B i^i D))

Syntax hints:   -> wi 3   = wceq 958   i^i cin 2049

This theorem is referenced by:  oev2 4168  mapdom2lem 4499  blin 7849  isps 8641  chocint 9413  cmbr3t 9546  pjoml3t 9550  fh1t 9556  erdisj2 10437

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054

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