Theorem sbrimvw 2103

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2317 based on fewer axioms, but with more disjoint variable conditions. (Contributed by Wolf Lammen, 29-Jan-2024.) Remove DV condition. (Revised by Wolf Lammen, 5-Jun-2026.)

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem sbrimvw

Step	Hyp	Ref	Expression
1		sbv 2100	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
2		sbi1 2083	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
3	1, 2	biimtrrid 245	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → (𝜑 → [𝑦 / 𝑥]𝜓))
4		sbv 2100	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜑)
5		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	sbimi 2086	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
7	4, 6	sylbir 237	. . 3 ⊢ (¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓))
8		ax-1 6	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
9	8	sbimi 2086	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓))
10	7, 9	ja 187	. 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
11	3, 10	impbii 211	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075

This theorem is referenced by:  sbiedvw  2108  cbvralsvw  3292