Theorem sbcid 3755

Description: An identity theorem for substitution. See sbid 2290. (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 3742	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑)
2		sbid 2290	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	1, 2	bitr3i 280	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2098  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3739

This theorem is used by:  csbid  3859  snfil  24188  ex-natded9.26  31025  bnj605  35583  dedths  40112  frege93  45049  or2expropbilem1  48195