Theorem u4lembi 706

Description: Non-tollens implication and biconditional.

Assertion

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

Proof of Theorem u4lembi

Step	Hyp	Ref	Expression
1		ud4lem1a 559	. 2 ((a →4 b) ∩ (b →4 a)) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		dfb 86	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
3	2	ax-r1 34	. 2 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ≡ b)
4	1, 3	ax-r2 35	1 ((a →4 b) ∩ (b →4 a)) = (a ≡ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₄ wi4 16

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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