Theorem sb2 1647

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

Assertion

Ref	Expression
sb2	⊢ (∀x(x = y → φ) → [y / x]φ)

Proof of Theorem sb2

Step	Hyp	Ref	Expression
1		ax-4 1397	. 2 ⊢ (∀x(x = y → φ) → (x = y → φ))
2		equs4 1610	. 2 ⊢ (∀x(x = y → φ) → ∃x(x = y ∧ φ))
3		df-sb 1643	. 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))
4	1, 2, 3	sylanbrc 394	1 ⊢ (∀x(x = y → φ) → [y / x]φ)

Syntax hints:   → 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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  stdpc4  1655  equsb1  1665  equsb2  1666  sbiedh  1667  sb6f  1681  hbsb2a  1684  hbsb2e  1685  sbcof2  1688  sb3  1709  sb4b  1712  sb4bor  1713  hbsb2  1714  nfsb2or  1715  sb6rf  1730  sbi1v  1768  sbalyz  1872  iota4  4828