Theorem axc11nlemOLD2 1988

Description: Lemma for axc11n 2307. Change bound variable in an equality. Obsolete as of 29-Mar-2021. Use aev 1983 instead. (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.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 14-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
axc11nlemOLD2.1	⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))

Assertion

Ref	Expression
axc11nlemOLD2	⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axc11nlemOLD2

Step	Hyp	Ref	Expression
1		cbvaev 1979	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)
2		equequ2 1953	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑧))
3	2	biimprd 238	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥))
4	3	al2imi 1743	. . 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		spaev 1978	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧)
7		axc11nlemOLD2.1	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
8	6, 7	syl5com 31	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
9	8	con1d 139	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
10	5, 9	pm2.61d 170	1 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  aevlemOLD  1989  axc11nOLDOLD  2309