Theorem axc11nlemOLD 2146

Description: Obsolete proof of axc11nlemOLD2 1975 as of 14-Mar-2021. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
axc11nlemOLD.1	⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))

Assertion

Ref	Expression
axc11nlemOLD	⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axc11nlemOLD

Step	Hyp	Ref	Expression
1		cbvaev 1966	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)
2		equequ2 1940	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑧))
3	2	biimprd 237	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥))
4	3	al2imi 1733	. . 3 ⊢ (∀𝑦 𝑥 = 𝑧 → (∀𝑦 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
5	1, 4	syl5com 31	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
6		axc11nlemOLD.1	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
7	6	spsd 2045	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
8	7	com12 32	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
9	8	con1d 138	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
10	5, 9	pm2.61d 169	1 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)