Theorem sb4e 1924

Description: One direction of a simplified definition of substitution that unlike sb4 2053 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
sb4e	⊢ ([y / x]φ → ∀x(x = y → ∃yφ))

Proof of Theorem sb4e

Step	Hyp	Ref	Expression
1		sb1 1651	. 2 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))
2		equs5e 1888	. 2 ⊢ (∃x(x = y ∧ φ) → ∀x(x = y → ∃yφ))
3	1, 2	syl 15	1 ⊢ ([y / x]φ → ∀x(x = y → ∃yφ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  [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-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  hbsb2e  2042