Theorem sbf 2271

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 2089. (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 2270	. 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206  Ⅎwnf 1783  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784  df-sb 2066

This theorem is referenced by:  sbf2  2272  sbh  2273  nfs1f  2275  sblim  2305  sbrbif  2310  sbiev  2313  sb8f  2352  sb6x  2463  sbequ5  2464  sbequ6  2465  sb2ae  2495  sbie  2501  sbid2  2507  sbabel  2925  sbhypf  3513  nfcdeq  3751  mo5f  32425  suppss2f  32569  fmptdF  32587  disjdsct  32633  esumpfinvalf  34073  bj-sbf3  36834  bj-sbf4  36835  ellimcabssub0  45622  2reu8i  47118  ichf  47455