Theorem ax12vOLDOLD 2051

Description: Obsolete proof of ax12v 2048 as of 7-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2019 and ax-13 2246. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax12vOLDOLD	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12vOLDOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1953	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
2	1	biimprd 238	. . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
3		ax-5 1839	. . . . 5 ⊢ (𝜑 → ∀𝑧𝜑)
4		ax-12 2047	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
5	3, 4	syl5 34	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
6	2	imim1d 82	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑)))
7	6	alimdv 1845	. . . 4 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	5, 7	syl9r 78	. . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	2, 8	syld 47	. 2 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
10		ax6ev 1890	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
11	9, 10	exlimiiv 1859	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → 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  ax-12 2047

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

This theorem is referenced by: (None)