Theorem isfildlem 22916

Description: Lemma for isfild 22917. (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 3440	. . 3 ⊢ (𝐵 ∈ 𝐹 → 𝐵 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∈ 𝐹 → 𝐵 ∈ V))
3		isfild.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ssexg 5242	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐵 ∈ V)
5	4	expcom 413	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ V))
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ V))
7	6	adantrd 491	. 2 ⊢ (𝜑 → ((𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓) → 𝐵 ∈ V))
8		eleq1 2826	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐹 ↔ 𝐵 ∈ 𝐹))
9		sseq1 3942	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
10		dfsbcq 3713	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ([𝑦 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
11	9, 10	anbi12d 630	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))
12	8, 11	bibi12d 345	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)) ↔ (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))))
13	12	imbi2d 340	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓))) ↔ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))))
14		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥𝜑
15		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐹
16		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
17		nfsbc1v 3731	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓
18	16, 17	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)
19	15, 18	nfbi 1907	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓))
20	14, 19	nfim 1900	. . . . 5 ⊢ Ⅎ𝑥(𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
21		eleq1 2826	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐹 ↔ 𝑦 ∈ 𝐹))
22		sseq1 3942	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
23		sbceq1a 3722	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
24	22, 23	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝐴 ∧ 𝜓) ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
25	21, 24	bibi12d 345	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)) ↔ (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓))))
26	25	imbi2d 340	. . . . 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 2236	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
29	13, 28	vtoclg 3495	. . 3 ⊢ (𝐵 ∈ V → (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))))
30	29	com12 32	. 2 ⊢ (𝜑 → (𝐵 ∈ V → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓))))
31	2, 7, 30	pm5.21ndd 380	1 ⊢ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  [wsbc 3711   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-sbc 3712  df-in 3890  df-ss 3900

This theorem is referenced by:  isfild  22917