Theorem nfsb4 2534

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (inference associated with nfsb4t 2533). Theorem nfsb 2557 replaces the distinctor antecedent with a disjoint variable condition. See nfsbv 2365 for a weaker version of nfsb 2557 not requiring ax-13 2406. (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 2406. Use nfsbv 2365 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
nfsb4.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsb4	⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb4

Step	Hyp	Ref	Expression
1		nfsb4t 2533	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2		nfsb4.1	. 2 ⊢ Ⅎ𝑧𝜑
3	1, 2	mpg 1830	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  Ⅎwnf 1816  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sbco2  2545