Theorem sb8e 2093

Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	⊢ Ⅎyφ

Assertion

Ref	Expression
sb8e	⊢ (∃xφ ↔ ∃y[y / x]φ)

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb5rf.1	. . . . . 6 ⊢ Ⅎyφ
2	1	nfn 1793	. . . . 5 ⊢ Ⅎy ¬ φ
3	2	sb8 2092	. . . 4 ⊢ (∀x ¬ φ ↔ ∀y[y / x] ¬ φ)
4		sbn 2062	. . . . 5 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
5	4	albii 1566	. . . 4 ⊢ (∀y[y / x] ¬ φ ↔ ∀y ¬ [y / x]φ)
6	3, 5	bitri 240	. . 3 ⊢ (∀x ¬ φ ↔ ∀y ¬ [y / x]φ)
7	6	notbii 287	. 2 ⊢ (¬ ∀x ¬ φ ↔ ¬ ∀y ¬ [y / x]φ)
8		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
9		df-ex 1542	. 2 ⊢ (∃y[y / x]φ ↔ ¬ ∀y ¬ [y / x]φ)
10	7, 8, 9	3bitr4i 268	1 ⊢ (∃xφ ↔ ∃y[y / x]φ)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [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:  exsbOLD  2131  sb8mo  2223