Theorem ml2i 1125

Description: Inference version of modular law. (Contributed by NM, 1-Apr-2012.)

Hypothesis

Ref	Expression
mli.1	c ≤ a

Assertion

Ref	Expression
ml2i	(c ∪ (b ∩ a)) = ((c ∪ b) ∩ a)

Proof of Theorem ml2i

Step	Hyp	Ref	Expression
1		ml 1123	. 2 (c ∪ (b ∩ (c ∪ a))) = ((c ∪ b) ∩ (c ∪ a))
2		mli.1	. . . . 5 c ≤ a
3	2	df-le2 131	. . . 4 (c ∪ a) = a
4	3	lan 77	. . 3 (b ∩ (c ∪ a)) = (b ∩ a)
5	4	lor 70	. 2 (c ∪ (b ∩ (c ∪ a))) = (c ∪ (b ∩ a))
6	3	lan 77	. 2 ((c ∪ b) ∩ (c ∪ a)) = ((c ∪ b) ∩ a)
7	1, 5, 6	3tr2 64	1 (c ∪ (b ∩ a)) = ((c ∪ b) ∩ a)

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1122

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by:  mli  1126  l42modlem1  1149  dp53lemb  1164  dp35lemb  1176  dp41lemd  1186  dp32  1196  xdp41  1198  xdp53  1200  xxdp41  1201  xxdp53  1203  xdp45lem  1204  xdp43lem  1205  xdp45  1206  xdp43  1207  3dp43  1208