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Theorem sbrimvw 2095
Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2301 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](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbrimvw
StepHypRef Expression
1 sb6 2089 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 bi2.04 389 . . . 4 ((𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝑥 = 𝑦 → (𝜑𝜓)))
32albii 1822 . . 3 (∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
4 19.21v 1943 . . 3 (∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
51, 3, 43bitr2i 299 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
6 sb6 2089 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
76imbi2i 336 . 2 ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
85, 7bitr4i 278 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  [wsb 2068
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 1914  ax-6 1972  ax-7 2012
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069
This theorem is referenced by:  sbiedvw  2097
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