Theorem bj-sbidmOLD 34176

Description: Obsolete proof of sbidm 2552 temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-sbidmOLD	⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem bj-sbidmOLD

Step	Hyp	Ref	Expression
1		equsb2 2531	. . 3 ⊢ [𝑦 / 𝑥]𝑦 = 𝑥
2		sbequ12r 2254	. . . 4 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	2	sbimi 2079	. . 3 ⊢ ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑 ↔ 𝜑))
4	1, 3	ax-mp 5	. 2 ⊢ [𝑦 / 𝑥]([𝑦 / 𝑥]𝜑 ↔ 𝜑)
5		sbbi 2317	. 2 ⊢ ([𝑦 / 𝑥]([𝑦 / 𝑥]𝜑 ↔ 𝜑) ↔ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
6	4, 5	mpbi 232	1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)