Theorem cancel 874

Description: Cancellation law eliminating →₁ consequent.

Hypothesis

Ref	Expression
cancel.1	((d ∪ (a →1 c)) →1 c) = ((d ∪ (b →1 c)) →1 c)

Assertion

Ref	Expression
cancel	(d ∪ (a →1 c)) = (d ∪ (b →1 c))

Proof of Theorem cancel

Step	Hyp	Ref	Expression
1		cancel.1	. . 3 ((d ∪ (a →1 c)) →1 c) = ((d ∪ (b →1 c)) →1 c)
2	1	cancellem 873	. 2 (d ∪ (a →1 c)) ≤ (d ∪ (b →1 c))
3	1	ax-r1 34	. . 3 ((d ∪ (b →1 c)) →1 c) = ((d ∪ (a →1 c)) →1 c)
4	3	cancellem 873	. 2 (d ∪ (b →1 c)) ≤ (d ∪ (a →1 c))
5	2, 4	lebi 137	1 (d ∪ (a →1 c)) = (d ∪ (b →1 c))

Syntax hints:   = wb 1   ∪ wo 6   →₁ wi1 13

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

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