Theorem sbf 1770

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sbf.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbf	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Proof of Theorem sbf

Step	Hyp	Ref	Expression
1		sbf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1512	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	sbh 1769	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 104  Ⅎwnf 1453  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbf2  1771  sbequ5  1775  sbequ6  1776  sbt  1777  sblim  1950  moimv  2085  moanim  2093  sbabel  2339  nfcdeq  2952  oprcl  3789