Theorem u4lembi 724

Description: Non-tollens implication and biconditional. (Contributed by NM, 17-Jan-1998.)

Assertion

Ref	Expression
u4lembi	((a →4 b) ∩ (b →4 a)) = (a ≡ b)

Proof of Theorem u4lembi

Step	Hyp	Ref	Expression
1		ud4lem1a 577	. 2 ((a →4 b) ∩ (b →4 a)) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
3	2	ax-r1 35	. 2 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ≡ b)
4	1, 3	ax-r2 36	1 ((a →4 b) ∩ (b →4 a)) = (a ≡ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₄ wi4 15

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-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)