Theorem sbrimvw 2100

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2310 and sbrimv 2311 based on fewer axioms, but with more disjoint variable conditions. Based on an idea of Gino Giotto. (Contributed by Wolf Lammen, 29-Jan-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

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

Proof of Theorem sbrimvw

Step	Hyp	Ref	Expression
1		19.21v 1941	. 2 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
2	1	sbrimvlem 2099	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 209  [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

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

This theorem is referenced by:  sbiedvw  2102