Theorem sbiedvw 2100

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbievw 2098). Version of sbied 2505 and sbiedv 2506 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by GG, 29-Jan-2024.)

Hypothesis

Ref	Expression
sbiedvw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥   𝑥,𝑦

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

Proof of Theorem sbiedvw

Step	Hyp	Ref	Expression
1		sbrimvw 2096	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
2		sbiedvw.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
3	2	expcom 413	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
4	3	pm5.74d 273	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
5	4	sbievw 2098	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
6	1, 5	bitr3i 277	. 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → 𝜒))
7	6	pm5.74ri 272	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  [wsb 2067

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 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068

This theorem is referenced by:  2sbievw  2101  iscatd2  17597  bj-elabd2ALT  36980