Theorem wl-speqv 37482

Description: Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2182 is provable from Tarski's FOL and ax13v 2376 only. Note that this reverts the implication in ax13lem1 2377, so in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)

Assertion

Ref	Expression
wl-speqv	⊢ (¬ 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-speqv

Step	Hyp	Ref	Expression
1		19.2 1975	. 2 ⊢ (∀𝑥 𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)
2		ax13lem2 2379	. 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
3	1, 2	syl5 34	1 ⊢ (¬ 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)