Theorem ax12v2 2217

Description: It is possible to remove any restriction on 𝜑 in ax12v 2216. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2216 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-10 2178 and ax-13 2406. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)

Assertion

Ref	Expression
ax12v2	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12v2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equtrr 2045	. . 3 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
2		ax12v 2216	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
3	1	imim1d 83	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑)))
4	3	alimdv 1939	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	2, 4	syl9r 79	. . 3 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
6	1, 5	syld 48	. 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
7		ax6evr 2038	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
8	6, 7	exlimiiv 1954	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  ax12ev2  2218  sbalexOLD  2281  equs5av  2314  in-ax8  36592  ss-ax8  36593  wl-lem-exsb  38076  wl-lem-moexsb  38078