Theorem sbh 2310

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-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	nf5i 2184	. 2 ⊢ Ⅎ𝑥𝜑
3	2	sbf 2308	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by: (None)