Theorem sbh 2027

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sbh.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
sbh	⊢ ([y / x]φ ↔ φ)

Proof of Theorem sbh

Step	Hyp	Ref	Expression
1		sbh.1	. . 3 ⊢ (φ → ∀xφ)
2	1	nfi 1551	. 2 ⊢ Ⅎxφ
3	2	sbf 2026	1 ⊢ ([y / x]φ ↔ φ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  [wsb 1648

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

This theorem is used by: (None)