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Theorem sb8f 2354
Description: Substitution of variable in universal quantifier. Version of sb8 2520 with a disjoint variable condition, not requiring ax-10 2139 or ax-13 2375. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2139. (Revised by SN, 5-Dec-2024.)
Hypothesis
Ref Expression
sb8f.nf 𝑦𝜑
Assertion
Ref Expression
sb8f (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8f
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 sb6 2083 . . . 4 ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥𝜑))
5 sb8f.nf . . . . 5 𝑦𝜑
65sbf 2269 . . . 4 ([𝑥 / 𝑦]𝜑𝜑)
7 equcom 2015 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
87imbi1i 349 . . . . 5 ((𝑦 = 𝑥𝜑) ↔ (𝑥 = 𝑦𝜑))
98albii 1816 . . . 4 (∀𝑦(𝑦 = 𝑥𝜑) ↔ ∀𝑦(𝑥 = 𝑦𝜑))
104, 6, 93bitr3ri 302 . . 3 (∀𝑦(𝑥 = 𝑦𝜑) ↔ 𝜑)
1110albii 1816 . 2 (∀𝑥𝑦(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜑)
122, 3, 113bitrri 298 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wnf 1780  [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  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  mo5f  32517  ax11-pm2  36819  bj-nfcf  36906
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