Theorem axc16i 2458

Description: Inference with axc16 2262 as its conclusion. Usage of axc16 2262 is preferred since it requires fewer axioms. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
axc16i.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
axc16i.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
axc16i	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem axc16i

Step	Hyp	Ref	Expression
1		nfv 1915	. . 3 ⊢ Ⅎ𝑧 𝑥 = 𝑦
2		nfv 1915	. . 3 ⊢ Ⅎ𝑥 𝑧 = 𝑦
3		ax7 2023	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
4	1, 2, 3	cbv3 2415	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
5		ax7 2023	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
6	5	spimvw 2002	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → 𝑥 = 𝑦)
7		equcomi 2024	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
8		equcomi 2024	. . . . . . 7 ⊢ (𝑧 = 𝑦 → 𝑦 = 𝑧)
9		ax7 2023	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑥 → 𝑧 = 𝑥))
10	8, 9	syl 17	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑦 = 𝑥 → 𝑧 = 𝑥))
11	7, 10	syl5com 31	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
12	11	alimdv 1917	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
13	6, 12	mpcom 38	. . 3 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
14		equcomi 2024	. . . 4 ⊢ (𝑧 = 𝑥 → 𝑥 = 𝑧)
15	14	alimi 1812	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → ∀𝑧 𝑥 = 𝑧)
16	13, 15	syl 17	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
17		axc16i.1	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
18	17	biimpcd 251	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑧 → 𝜓))
19	18	alimdv 1917	. . 3 ⊢ (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧𝜓))
20		axc16i.2	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
21	20	nf5i 2150	. . . 4 ⊢ Ⅎ𝑥𝜓
22		nfv 1915	. . . 4 ⊢ Ⅎ𝑧𝜑
23	17	biimprd 250	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜓 → 𝜑))
24	14, 23	syl 17	. . . 4 ⊢ (𝑧 = 𝑥 → (𝜓 → 𝜑))
25	21, 22, 24	cbv3 2415	. . 3 ⊢ (∀𝑧𝜓 → ∀𝑥𝜑)
26	19, 25	syl6com 37	. 2 ⊢ (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
27	4, 16, 26	3syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by:  axc16ALT  2528