Theorem sbf 2026

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	. 2 ⊢ Ⅎxφ
2		sbft 2025	. 2 ⊢ (Ⅎxφ → ([y / x]φ ↔ φ))
3	1, 2	ax-mp 5	1 ⊢ ([y / x]φ ↔ φ)

Syntax hints:   ↔ wb 176  Ⅎwnf 1544  [wsb 1648

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbh  2027  sbf2  2028  sb6x  2029  nfs1f  2030  sbequ5  2031  sbequ6  2032  sbt  2033  sbrim  2067  sblim  2068  sbrbif  2074  sbid2  2084  sbabel  2516