ml2i - Quantum Logic Explorer

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Theorem ml2i 1123

Description: Inference version of modular law.

**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 1121	. 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)

Colors of variables: term

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 1120

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 1124 l42modlem1 1147 dp53lemb 1162 dp35lemb 1174 dp41lemd 1184 dp32 1194 xdp41 1196 xdp53 1198 xxdp41 1199 xxdp53 1201 xdp45lem 1202 xdp43lem 1203 xdp45 1204 xdp43 1205 3dp43 1206