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Theorem sbrimvlem 2097
Description: Common proof template for sbrimvw 2098 and sbrimv 2306. 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 2091 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 bi2.04 392 . . . 4 ((𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝑥 = 𝑦 → (𝜑𝜓)))
32albii 1827 . . 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 2091 . . 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 1541  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071
This theorem is referenced by:  sbrimvw  2098  sbrimv  2306
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