Theorem eqvin 932

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109.

Assertion

Ref	Expression
eqvin	⊢ (x = y ↔ ∃z(x = z ∧ z = y))

Distinct variable group(s):   x,z   y,z

Proof of Theorem eqvin

Step	Hyp	Ref	Expression
1		eqvin.l1 851	. 2 ⊢ (x = y → ∃z(x = z ∧ z = y))
2		eqvin.l2 931	. 2 ⊢ (∃z(x = z ∧ z = y) → x = y)
3	1, 2	impbi 139	1 ⊢ (x = y ↔ ∃z(x = z ∧ z = y))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797

This theorem is referenced by:  eqvinc 1407

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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