Theorem sb1 1815

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 1812	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	simprbi 275	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1541  [wsb 1811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-sb 1812

This theorem is referenced by:  sbh  1825  sbiedh  1836  sb4a  1850  sb4e  1854  sbcof2  1859  sb4  1881  sb4or  1882  spsbe  1891  sbidm  1900  sb5rf  1901  bj-sbimedh  16543