Theorem sbcid 2948

Description: An identity theorem for substitution. See sbid 1751. (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 2937	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑)
2		sbid 1751	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	1, 2	bitr3i 185	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 104  [wsb 1739  [wsbc 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-sbc 2934

This theorem is referenced by:  csbid  3035