Theorem sbf 1788

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 1530	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	sbh 1787	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 105  Ⅎwnf 1471  [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbf2  1789  sbequ5  1793  sbequ6  1794  sbt  1795  sblim  1969  moimv  2104  moanim  2112  sbabel  2359  nfcdeq  2974  oprcl  3817