Theorem wl-equsb4 33829

Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (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 2388	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2	1	ex 402	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
3		sbft 2496	. . 3 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
4	2, 3	syl6com 37	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧)))
5		sbequ12r 2279	. . . 4 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
6	5	equcoms 2119	. . 3 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
7	6	sps 2219	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
8	4, 7	pm2.61d2 174	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1651  Ⅎwnf 1879  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213  ax-13 2377

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065

This theorem is referenced by: (None)