Theorem sbrimv 2311

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2310 not depending on ax-10 2143, but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024.)

Hypothesis

Ref	Expression
sbrim.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbrimv

Step	Hyp	Ref	Expression
1		sbrim.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	19.21 2206	. 2 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
3	2	sbrimvlem 2099	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1786  [wsb 2070

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 1912  ax-6 1971  ax-7 2016  ax-12 2176

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-nf 1787  df-sb 2071

This theorem is referenced by:  sbiedw  2324