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Theorem nfsbvOLD 2330
Description: Obsolete version of nfsbv 2329 as of 25-Oct-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nfsbv.nf 𝑧𝜑
Assertion
Ref Expression
nfsbvOLD 𝑧[𝑦 / 𝑥]𝜑
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbvOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2063 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑)))
2 nfv 1912 . . . 4 𝑧 𝑤 = 𝑦
3 nfv 1912 . . . . . 6 𝑧 𝑥 = 𝑤
4 nfsbv.nf . . . . . 6 𝑧𝜑
53, 4nfim 1894 . . . . 5 𝑧(𝑥 = 𝑤𝜑)
65nfal 2322 . . . 4 𝑧𝑥(𝑥 = 𝑤𝜑)
72, 6nfim 1894 . . 3 𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑))
87nfal 2322 . 2 𝑧𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤𝜑))
91, 8nfxfr 1850 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  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-10 2139  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: (None)
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