Theorem equvin 1277

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

Assertion

Ref	Expression
equvin	⊢ (x = y ↔ ∃z(x = z ⋀ z = y))

Distinct variable groups:   x,z   y,z

Proof of Theorem equvin

Step	Hyp	Ref	Expression
1		equvini 1170	. 2 ⊢ (x = y → ∃z(x = z ⋀ z = y))
2		ax-17 973	. . 3 ⊢ (x = y → ∀z x = y)
3		equtr 1133	. . . 4 ⊢ (x = z → (z = y → x = y))
4	3	imp 350	. . 3 ⊢ ((x = z ⋀ z = y) → x = y)
5	2, 4	19.23ai 1066	. 2 ⊢ (∃z(x = z ⋀ z = y) → x = y)
6	1, 5	impbi 157	1 ⊢ (x = y ↔ ∃z(x = z ⋀ z = y))

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 958  ∃wex 982

This theorem is referenced by:  eqvinc 1886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983

Copyright terms: Public domain