Theorem sbcid 3787

Description: An identity theorem for substitution. See sbid 2239. (Contributed by Mario Carneiro, 18-Feb-2017.)

Assertion

Ref	Expression
sbcid	⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Proof of Theorem sbcid

Step	Hyp	Ref	Expression
1		sbsbc 3774	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑)
2		sbid 2239	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	1, 2	bitr3i 277	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  [wsb 2059  [wsbc 3770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-sbc 3771

This theorem is referenced by:  csbid  3899  snfil  23712  ex-natded9.26  30166  bnj605  34436  dedths  38335  frege93  43256  or2expropbilem1  46287