Theorem sb4 2053

Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sb4	⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))

Proof of Theorem sb4

Step	Hyp	Ref	Expression
1		sb1 1651	. 2 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))
2		equs5 1996	. 2 ⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))
3	1, 2	syl5 28	1 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀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:  sb4b  2054  dfsb2  2055  hbsb2  2057  sbn  2062  sbi1  2063  sbal1  2126