Theorem sbidm 2507

Description: An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
sbidm	⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbidm

Step	Hyp	Ref	Expression
1		sbcom3 2503	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
2		sbid 2245	. . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	2	sbbii 2077	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitr3i 276	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2065

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 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169  ax-13 2369

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-sb 2066

This theorem is referenced by: (None)