Theorem sbiedw 2310

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2309). Version of sbied 2507 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
sbiedw.1	⊢ Ⅎ𝑥𝜑
sbiedw.2	⊢ (𝜑 → Ⅎ𝑥𝜒)
sbiedw.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
sbiedw	⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem sbiedw

Step	Hyp	Ref	Expression
1		sbiedw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	sbrim 2301	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
3		sbiedw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	1, 3	nfim1 2192	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜒)
5		sbiedw.3	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	5	com12 32	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
7	6	pm5.74d 272	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
8	4, 7	sbiev 2309	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
9	2, 8	bitr3i 276	. 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → 𝜒))
10	9	pm5.74ri 271	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1786  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by:  csbie2df  4374