Theorem eqan 816

Description: A transitive law for equality.

Assertion

Ref	Expression
eqan	⊢ ((x = z ∧ y = z) → x = y)

Proof of Theorem eqan

Step	Hyp	Ref	Expression
1		eqt 814	. . 3 ⊢ (x = z → (z = y → x = y))
2		eqcom 811	. . 3 ⊢ (y = z → z = y)
3	1, 2	syl5 22	. 2 ⊢ (x = z → (y = z → x = y))
4	3	imp 277	1 ⊢ ((x = z ∧ y = z) → x = y)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797

This theorem is referenced by:  mo 1020

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-8 798  ax-9 799  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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