Theorem combi 485

Description: Commutation theorem for Sasaki implication. (Contributed by NM, 25-Oct-1998.)

Assertion

Ref	Expression
combi	a C (a ≡ b)

Proof of Theorem combi

Step	Hyp	Ref	Expression
1		comanr1 464	. . 3 a C (a ∩ b)
2		comanr1 464	. . . 4 a⊥ C (a⊥ ∩ b⊥ )
3	2	comcom6 459	. . 3 a C (a⊥ ∩ b⊥ )
4	1, 3	com2or 483	. 2 a C ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
5		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
6	5	ax-r1 35	. 2 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ≡ b)
7	4, 6	cbtr 182	1 a C (a ≡ b)

Syntax hints:   C wc 3  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7

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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  ublemc1  728