Theorem frind 4417

Description: Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)

Hypotheses

Ref	Expression
frind.sb	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
frind.ind	⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑))
frind.fr	⊢ (𝜒 → 𝑅 Fr 𝐴)
frind.a	⊢ (𝜒 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
frind	⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜒,𝑥   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem frind

Dummy variables 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frind.ind	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑))
2	1	ralrimiva 2581	. . . . . . 7 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑))
3		nfv 1552	. . . . . . . 8 ⊢ Ⅎ𝑧(∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑)
4		nfv 1552	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓)
5		nfs1v 1968	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfim 1596	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓) → [𝑧 / 𝑥]𝜑)
7		breq2 4063	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
8	7	imbi1d 231	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑦𝑅𝑥 → 𝜓) ↔ (𝑦𝑅𝑧 → 𝜓)))
9	8	ralbidv 2508	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓)))
10		sbequ12 1795	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
11	9, 10	imbi12d 234	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓) → [𝑧 / 𝑥]𝜑)))
12	3, 6, 11	cbvral 2738	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑) ↔ ∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓) → [𝑧 / 𝑥]𝜑))
13	2, 12	sylib 122	. . . . . 6 ⊢ (𝜒 → ∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓) → [𝑧 / 𝑥]𝜑))
14		frind.sb	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
15	14	elrab3 2937	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝜓))
16	15	imbi2d 230	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → ((𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ (𝑦𝑅𝑧 → 𝜓)))
17	16	ralbiia 2522	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓))
18	17	a1i 9	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓)))
19		nfcv 2350	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
20		nfcv 2350	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
21	19, 20, 5, 10	elrabf 2934	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
22	21	baib 921	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑))
23	18, 22	imbi12d 234	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓) → [𝑧 / 𝑥]𝜑)))
24	23	ralbiia 2522	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) ↔ ∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝜓) → [𝑧 / 𝑥]𝜑))
25	13, 24	sylibr 134	. . . . 5 ⊢ (𝜒 → ∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}))
26		frind.fr	. . . . . . . 8 ⊢ (𝜒 → 𝑅 Fr 𝐴)
27		df-frind 4397	. . . . . . . 8 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
28	26, 27	sylib 122	. . . . . . 7 ⊢ (𝜒 → ∀𝑠 FrFor 𝑅𝐴𝑠)
29		frind.a	. . . . . . . 8 ⊢ (𝜒 → 𝐴 ∈ 𝑉)
30		rabexg 4203	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
31		frforeq3 4412	. . . . . . . . 9 ⊢ (𝑠 = {𝑥 ∈ 𝐴 ∣ 𝜑} → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑅𝐴{𝑥 ∈ 𝐴 ∣ 𝜑}))
32	31	spcgv 2867	. . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V → (∀𝑠 FrFor 𝑅𝐴𝑠 → FrFor 𝑅𝐴{𝑥 ∈ 𝐴 ∣ 𝜑}))
33	29, 30, 32	3syl 17	. . . . . . 7 ⊢ (𝜒 → (∀𝑠 FrFor 𝑅𝐴𝑠 → FrFor 𝑅𝐴{𝑥 ∈ 𝐴 ∣ 𝜑}))
34	28, 33	mpd 13	. . . . . 6 ⊢ (𝜒 → FrFor 𝑅𝐴{𝑥 ∈ 𝐴 ∣ 𝜑})
35		df-frfor 4396	. . . . . 6 ⊢ ( FrFor 𝑅𝐴{𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
36	34, 35	sylib 122	. . . . 5 ⊢ (𝜒 → (∀𝑧 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑧 → 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑}))
37	25, 36	mpd 13	. . . 4 ⊢ (𝜒 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑})
38		ssrab 3279	. . . 4 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑))
39	37, 38	sylib 122	. . 3 ⊢ (𝜒 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑))
40	39	simprd 114	. 2 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 𝜑)
41	40	r19.21bi 2596	1 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371  [wsb 1786   ∈ wcel 2178  ∀wral 2486  {crab 2490  Vcvv 2776   ⊆ wss 3174   class class class wbr 4059   FrFor wfrfor 4392   Fr wfr 4393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-sep 4178

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-frfor 4396  df-frind 4397

This theorem is referenced by: (None)