Theorem sbcid 3015

Description: An identity theorem for substitution. See sbid 1798. (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 3003	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑)
2		sbid 1798	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	1, 2	bitr3i 186	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 105  [wsb 1786  [wsbc 2999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-sbc 3000

This theorem is referenced by:  csbid  3102