Theorem sb4b 2054

Description: Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Assertion

Ref	Expression
sb4b	⊢ (¬ ∀x x = y → ([y / x]φ ↔ ∀x(x = y → φ)))

Proof of Theorem sb4b

Step	Hyp	Ref	Expression
1		sb4 2053	. 2 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))
2		sb2 2023	. 2 ⊢ (∀x(x = y → φ) → [y / x]φ)
3	1, 2	impbid1 194	1 ⊢ (¬ ∀x x = y → ([y / x]φ ↔ ∀x(x = y → φ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  [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:  nfsb4t  2080  sbcom  2089  sbcom2  2114