Theorem spsbe 2075

Description: A specialization theorem. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
spsbe	⊢ ([y / x]φ → ∃xφ)

Proof of Theorem spsbe

Step	Hyp	Ref	Expression
1		stdpc4 2024	. . . 4 ⊢ (∀x ¬ φ → [y / x] ¬ φ)
2		sbn 2062	. . . 4 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
3	1, 2	sylib 188	. . 3 ⊢ (∀x ¬ φ → ¬ [y / x]φ)
4	3	con2i 112	. 2 ⊢ ([y / x]φ → ¬ ∀x ¬ φ)
5		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
6	4, 5	sylibr 203	1 ⊢ ([y / x]φ → ∃xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  [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)