Theorem nfsb4 2503

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable. Usage of this theorem is discouraged because it depends on ax-13 2371. Theorem nfsb 2526 replaces the distinctor with a disjoint variable condition. Visit also nfsbv 2329 for a weaker version of nfsb 2526 not requiring ax-13 2371. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (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 2502	. 2 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2		nfsb4.1	. 2 ⊢ Ⅎ𝑧𝜑
3	1, 2	mpg 1805	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541  Ⅎwnf 1791  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by:  sbco2  2514  nfsbOLD  2527