Theorem isfildlem 23982

Description: Lemma for isfild 23983. (Contributed by Mario Carneiro, 1-Dec-2013.)

Hypotheses

Ref	Expression
isfild.1	⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))
isfild.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
isfildlem	⊢ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem isfildlem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3484	. . 3 ⊢ (𝐵 ∈ 𝐹 → 𝐵 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∈ 𝐹 → 𝐵 ∈ V))
3		isfild.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssexg 5294	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐵 ∈ V)
5	4	expcom 418	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ V))
6	3, 5	syl 18	. . 3 ⊢ (𝜑 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ V))
7	6	adantrd 496	. 2 ⊢ (𝜑 → ((𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓) → 𝐵 ∈ V))
8		eleq1 2857	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐹 ↔ 𝐵 ∈ 𝐹))
9		sseq1 3970	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
10		dfsbcq 3755	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
11	9, 10	anbi12d 643	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))
12	8, 11	bibi12d 348	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)) ↔ (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))))
13	12	imbi2d 343	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓))) ↔ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))))
14		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑥𝜑
15		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐹
16		nfv 1941	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
17		nfsbc1v 3773	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓
18	16, 17	nfan 1926	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)
19	15, 18	nfbi 1930	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓))
20	14, 19	nfim 1923	. . . . 5 ⊢ Ⅎ𝑥(𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
21		eleq1 2857	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐹 ↔ 𝑦 ∈ 𝐹))
22		sseq1 3970	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
23		sbceq1a 3764	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
24	22, 23	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝐴 ∧ 𝜓) ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
25	21, 24	bibi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)) ↔ (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓))))
26	25	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓))) ↔ (𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))))
27		isfild.1	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))
28	20, 26, 27	chvarfv 2282	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
29	13, 28	vtoclg 3531	. . 3 ⊢ (𝐵 ∈ V → (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))))
30	29	com12 33	. 2 ⊢ (𝜑 → (𝐵 ∈ V → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))))
31	2, 7, 30	pm5.21ndd 382	1 ⊢ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  [wsbc 3753   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-sbc 3754  df-in 3920  df-ss 3930

This theorem is referenced by:  isfild  23983