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Theorem sb8v 2353
Description: Substitution of variable in universal quantifier. Version of sb8f 2354 with a disjoint variable condition replacing the nonfree hypothesis 𝑦𝜑, not requiring ax-12 2175. (Contributed by SN, 5-Dec-2024.)
Assertion
Ref Expression
sb8v (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sb8v
StepHypRef Expression
1 sb6 2083 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
21albii 1816 . 2 (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥(𝑥 = 𝑦𝜑))
3 alcom 2157 . 2 (∀𝑦𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝑦(𝑥 = 𝑦𝜑))
4 equcom 2015 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
54imbi1i 349 . . . . 5 ((𝑥 = 𝑦𝜑) ↔ (𝑦 = 𝑥𝜑))
65albii 1816 . . . 4 (∀𝑦(𝑥 = 𝑦𝜑) ↔ ∀𝑦(𝑦 = 𝑥𝜑))
7 equsv 2000 . . . 4 (∀𝑦(𝑦 = 𝑥𝜑) ↔ 𝜑)
86, 7bitri 275 . . 3 (∀𝑦(𝑥 = 𝑦𝜑) ↔ 𝜑)
98albii 1816 . 2 (∀𝑥𝑦(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜑)
102, 3, 93bitrri 298 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-11 2155
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063
This theorem is referenced by:  sbnf2  2359  sb8eulem  2596  cbvralsvw  3315  abv  3490  abvALT  3491  wl-sb8eutv  37560  sbcalf  38101
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