Theorem sb1 1646

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

Assertion

Ref	Expression
sb1	⊢ ([y / x]φ → ∃x(x = y ∧ φ))

Proof of Theorem sb1

Step	Hyp	Ref	Expression
1		df-sb 1643	. 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))
2	1	simprbi 260	1 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))

Syntax hints:   → wi 4   ∧ wa 97  ∃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

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

This theorem is referenced by:  sbh  1656  sbiedh  1667  sb4a  1679  sb4e  1683  sbcof2  1688  sb4  1710  sb4or  1711  spsbe  1720  sbidm  1728  sb5rf  1729  bj-sbimedh  9226