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Theorem nfsb4 2494
Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (inference associated with nfsb4t 2493). Theorem nfsb 2517 replaces the distinctor antecedent with a disjoint variable condition. See nfsbv 2318 for a weaker version of nfsb 2517 not requiring ax-13 2366. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2366. Use nfsbv 2318 instead. (New usage is discouraged.)
Hypothesis
Ref Expression
nfsb4.1 𝑧𝜑
Assertion
Ref Expression
nfsb4 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb4
StepHypRef Expression
1 nfsb4t 2493 . 2 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2 nfsb4.1 . 2 𝑧𝜑
31, 2mpg 1792 1 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532  wnf 1778  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2164  ax-13 2366
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061
This theorem is referenced by:  sbco2  2505
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