Theorem sbrimv 2305

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2304 not depending on ax-10 2139, 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 2203	. 2 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
3	2	sbrimvlem 2095	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  sbiedw  2313