Theorem negantlem5 835

Description: Negated antecedent identity.

Hypothesis

Ref	Expression
negant.1	(a →1 c) = (b →1 c)

Assertion

Ref	Expression
negantlem5	(a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )

Proof of Theorem negantlem5

Step	Hyp	Ref	Expression
1		negant.1	. . 3 (a →1 c) = (b →1 c)
2	1	ran 71	. 2 ((a →1 c) ∩ c⊥ ) = ((b →1 c) ∩ c⊥ )
3		u1lemanb 597	. 2 ((a →1 c) ∩ c⊥ ) = (a⊥ ∩ c⊥ )
4		u1lemanb 597	. 2 ((b →1 c) ∩ c⊥ ) = (b⊥ ∩ c⊥ )
5	2, 3, 4	3tr2 61	1 (a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  negantlem6 836  negantlem7 837  negantlem9 841

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org