Theorem sbidm 2513

Description: An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 2509	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
2		sbid 2254	. . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	2	sbbii 2075	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitr3i 277	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by: (None)