Theorem sbf 2278

Description: Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv 2094. (Contributed by NM, 14-May-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 2277	. 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  Ⅎwnf 1785  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-sb 2069

This theorem is referenced by:  sbf2  2279  sbh  2280  nfs1f  2282  sblim  2312  sbrbif  2317  sbiev  2319  sb8f  2358  sb6x  2468  sbequ5  2469  sbequ6  2470  sb2ae  2500  sbie  2506  sbid2  2512  sbabel  2931  sbhypf  3490  nfcdeq  3723  mo5f  32558  suppss2f  32711  fmptdF  32729  disjdsct  32776  esumpfinvalf  34220  bj-sbf3  37146  bj-sbf4  37147  ellimcabssub0  46047  2reu8i  47561  ichf  47910