Theorem sb1 1651

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 1649	. 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))
2	1	simprbi 450	1 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541  [wsb 1648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-sb 1649

This theorem is used by:  sb4a  1923  sb4e  1924  sbft  2025  sbied  2036  sb4  2053  sbn  2062  sb5rf  2090