Theorem w2an 373

Description: Join both sides with conjunction. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
w2an.1	(a ≡ b) = 1
w2an.2	(c ≡ d) = 1

Assertion

Ref	Expression
w2an	((a ∩ c) ≡ (b ∩ d)) = 1

Proof of Theorem w2an

Step	Hyp	Ref	Expression
1		w2an.2	. . 3 (c ≡ d) = 1
2	1	wlan 370	. 2 ((a ∩ c) ≡ (a ∩ d)) = 1
3		w2an.1	. . 3 (a ≡ b) = 1
4	3	wran 369	. 2 ((a ∩ d) ≡ (b ∩ d)) = 1
5	2, 4	wr2 371	1 ((a ∩ c) ≡ (b ∩ d)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∩ wa 7  1wt 8

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  wcomd  418  wcom3ii  419  wcomcom5  420  wfh1  423  wfh3  425  wfh4  426  wddi4  1110  wdid0id5  1111  wdid0id1  1112  wdid0id2  1113