Theorem exsbOLD 2131

Description: An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
exsbOLD	⊢ (∃xφ ↔ ∃y∀x(x = y → φ))

Distinct variable groups:   x,y   φ,y

Allowed substitution hint:   φ(x)

Proof of Theorem exsbOLD

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎyφ
2	1	sb8e 2093	. 2 ⊢ (∃xφ ↔ ∃y[y / x]φ)
3		sb6 2099	. . 3 ⊢ ([y / x]φ ↔ ∀x(x = y → φ))
4	3	exbii 1582	. 2 ⊢ (∃y[y / x]φ ↔ ∃y∀x(x = y → φ))
5	2, 4	bitri 240	1 ⊢ (∃xφ ↔ ∃y∀x(x = y → φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)