Theorem hbsb2e 2042

Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
hbsb2e	⊢ ([y / x]φ → ∀x[y / x]∃yφ)

Proof of Theorem hbsb2e

Step	Hyp	Ref	Expression
1		sb4e 1924	. 2 ⊢ ([y / x]φ → ∀x(x = y → ∃yφ))
2		sb2 2023	. . 3 ⊢ (∀x(x = y → ∃yφ) → [y / x]∃yφ)
3	2	a5i 1789	. 2 ⊢ (∀x(x = y → ∃yφ) → ∀x[y / x]∃yφ)
4	1, 3	syl 15	1 ⊢ ([y / x]φ → ∀x[y / x]∃yφ)

Syntax hints:   → wi 4  ∀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-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)