Theorem wl-equsb4 38240

Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable condition. (Contributed by Wolf Lammen, 26-Jun-2019.)

Assertion

Ref	Expression
wl-equsb4	⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))

Proof of Theorem wl-equsb4

Step	Hyp	Ref	Expression
1		nfeqf 2412	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2	1	ex 417	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
3		sbft 2304	. . 3 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
4	2, 3	syl6com 38	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧)))
5		sbequ12r 2287	. . . 4 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
6	5	equcoms 2049	. . 3 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
7	6	sps 2220	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
8	4, 7	pm2.61d2 183	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1567  Ⅎwnf 1812  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096

This theorem is used by: (None)