Theorem sbf 1657

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 1409	. 2 ⊢ (φ → ∀xφ)
3	2	sbh 1656	1 ⊢ ([y / x]φ ↔ φ)

Syntax hints:   ↔ wb 98  Ⅎwnf 1346  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbf2  1658  sbequ5  1662  sbequ6  1663  sbt  1664  sblim  1828  moimv  1963  moanim  1971  sbabel  2200  nfcdeq  2755  oprcl  3564