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 2088. (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 2064

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 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by:  sbf2  2272  sbh  2273  nfs1f  2275  sbrimOLD  2305  sblim  2306  sbrbif  2311  sbiev  2314  sb8f  2355  sb6x  2468  sbequ5  2469  sbequ6  2470  sb2ae  2500  sbie  2506  sbid2  2512  sbabel  2931  sbhypf  3523  nfcdeq  3760  mo5f  32470  suppss2f  32616  fmptdF  32634  disjdsct  32680  esumpfinvalf  34107  bj-sbf3  36857  bj-sbf4  36858  ellimcabssub0  45646  2reu8i  47142  ichf  47464