Theorem aevlemOLD 1989

Description: Old proof of aevlem 1981. Obsolete as of 29-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2246, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
aevlemOLD	⊢ (∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥)

Distinct variable groups:   𝑧,𝑤   𝑥,𝑦

Proof of Theorem aevlemOLD

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvaev 1979	. 2 ⊢ (∀𝑧 𝑧 = 𝑤 → ∀𝑣 𝑣 = 𝑤)
2		ax5d 1840	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑣 → (𝑣 = 𝑤 → ∀𝑧 𝑣 = 𝑤))
3	2	axc11nlemOLD2 1988	. 2 ⊢ (∀𝑣 𝑣 = 𝑤 → ∀𝑧 𝑧 = 𝑣)
4		cbvaev 1979	. 2 ⊢ (∀𝑧 𝑧 = 𝑣 → ∀𝑥 𝑥 = 𝑣)
5		ax5d 1840	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑣 → ∀𝑦 𝑥 = 𝑣))
6	5	axc11nlemOLD2 1988	. 2 ⊢ (∀𝑥 𝑥 = 𝑣 → ∀𝑦 𝑦 = 𝑥)
7	1, 3, 4, 6	4syl 19	1 ⊢ (∀𝑧 𝑧 = 𝑤 → ∀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: (None)