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Theorem sbrimvlem 2098
 Description: Common proof template for sbrimvw 2099 and sbrimv 2310. The hypothesis is an instance of 19.21 2205. (Contributed by Wolf Lammen, 29-Jan-2024.)
Hypothesis
Ref Expression
sbrimvlem.1 (∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
Assertion
Ref Expression
sbrimvlem ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbrimvlem
StepHypRef Expression
1 sb6 2090 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 bi2.04 392 . . . 4 ((𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝑥 = 𝑦 → (𝜑𝜓)))
32albii 1821 . . 3 (∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
4 sbrimvlem.1 . . 3 (∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
51, 3, 43bitr2i 302 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
6 sb6 2090 . . 3 ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦𝜓))
76imbi2i 339 . 2 ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
85, 7bitr4i 281 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  sbrimvw  2099  sbrimv  2310
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