Theorem nfsb4 2504

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (inference associated with nfsb4t 2503). Theorem nfsb 2527 replaces the distinctor antecedent with a disjoint variable condition. See nfsbv 2328 for a weaker version of nfsb 2527 not requiring ax-13 2372. (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 2372. Use nfsbv 2328 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 2503	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2		nfsb4.1	. 2 ⊢ Ⅎ𝑧𝜑
3	1, 2	mpg 1801	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  sbco2  2515  nfsbOLD  2528