Theorem wl-19.2reqv 34929

Description: Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1981 is provable from Tarski's FOL and ax13v 2380 only. Note that in conjunction with 19.2 1981 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)

Assertion

Ref	Expression
wl-19.2reqv	⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-19.2reqv

Step	Hyp	Ref	Expression
1		ax13lem2 2383	. 2 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
2		ax13lem1 2381	. 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3	1, 2	syld 47	1 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

This theorem is referenced by: (None)