Theorem bj-sbieOLD 32020

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

Hypotheses

Ref	Expression
bj-sbieOLD.nf	⊢ Ⅎ𝑥𝜓
bj-sbieOLD.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem bj-sbieOLD

Step	Hyp	Ref	Expression
1		equsb1 2356	. . . 4 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
2		bj-sbieOLD.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbimi 1873	. . . 4 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	. . 3 ⊢ [𝑦 / 𝑥](𝜑 ↔ 𝜓)
5		sbbi 2389	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
6	4, 5	mpbi 219	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
7		bj-sbieOLD.nf	. . 3 ⊢ Ⅎ𝑥𝜓
8	7	sbf 2368	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
9	6, 8	bitri 263	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by: (None)