Theorem gstho 473

Description: "OR" version of Gudder-Schelp's Theorem.

Hypotheses

Ref	Expression
gstho.1	b C c
gstho.2	a C (b v c)

Assertion

Ref	Expression
gstho	(a v b) C c

Proof of Theorem gstho

Step	Hyp	Ref	Expression
1		anor3 82	. . . 4 (a_|_ ^ b_|_) = (a v b)_|_
2	1	ax-r1 34	. . 3 (a v b)_|_ = (a_|_ ^ b_|_)
3		gstho.1	. . . . 5 b C c
4	3	comcom4 437	. . . 4 b_|_ C c_|_
5		gstho.2	. . . . . 6 a C (b v c)
6	5	comcom4 437	. . . . 5 a_|_ C (b v c)_|_
7		anor3 82	. . . . . 6 (b_|_ ^ c_|_) = (b v c)_|_
8	7	ax-r1 34	. . . . 5 (b v c)_|_ = (b_|_ ^ c_|_)
9	6, 8	cbtr 174	. . . 4 a_|_ C (b_|_ ^ c_|_)
10	4, 9	gsth2 472	. . 3 (a_|_ ^ b_|_) C c_|_
11	2, 10	bctr 173	. 2 (a v b)_|_ C c_|_
12	11	comcom5 440	1 (a v b) C c

Syntax hints:   C wc 3  _|_wn 4   v wo 6   ^ wa 7

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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