Theorem sb4 1710

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 1646	. 2 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))
2		equs5 1707	. 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 97  ∀wal 1240  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643

This theorem is referenced by:  sb4b  1712  hbsb2  1714