Theorem csbie2df 4371

Description: Conversion of implicit substitution to explicit class substitution. This version of csbiedf 3859 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. Deduction form of csbie2 3870. (Contributed by AV, 29-Mar-2024.)

Hypotheses

Ref	Expression
csbie2df.p	⊢ Ⅎ𝑥𝜑
csbie2df.c	⊢ (𝜑 → Ⅎ𝑥𝐶)
csbie2df.d	⊢ (𝜑 → Ⅎ𝑥𝐷)
csbie2df.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
csbie2df.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
csbie2df.2	⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝐶 = 𝐷)

Assertion

Ref	Expression
csbie2df	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑦,𝐷   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbie2df

Step	Hyp	Ref	Expression
1		csbie2df.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		eqidd 2739	. . 3 ⊢ (𝜑 → 𝐷 = 𝐷)
3		dfsbcq 3713	. . . . . 6 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ [𝐴 / 𝑥]𝐵 = 𝐷))
4		sbceqg 4340	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐷))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐷))
6		csbie2df.d	. . . . . . . . . 10 ⊢ (𝜑 → Ⅎ𝑥𝐷)
7		csbtt 3845	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐷) → ⦋𝐴 / 𝑥⦌𝐷 = 𝐷)
8	6, 7	sylan2 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ⦋𝐴 / 𝑥⦌𝐷 = 𝐷)
9	8	eqeq2d 2749	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
10	5, 9	bitrd 278	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
11	1, 10	mpancom 684	. . . . . 6 ⊢ (𝜑 → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
12	3, 11	sylan9bb 509	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
13	12	pm5.74da 800	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 → [𝑦 / 𝑥]𝐵 = 𝐷) ↔ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)))
14		csbie2df.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝐶 = 𝐷)
15	14	eqeq1d 2740	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐷 = 𝐷))
16	15	expcom 413	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝜑 → (𝐶 = 𝐷 ↔ 𝐷 = 𝐷)))
17	16	pm5.74d 272	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝐶 = 𝐷) ↔ (𝜑 → 𝐷 = 𝐷)))
18		sbsbc 3715	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ [𝑦 / 𝑥]𝐵 = 𝐷)
19		csbie2df.p	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
20		csbie2df.c	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐶)
21	20, 6	nfeqd 2916	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝐶 = 𝐷)
22		csbie2df.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
23	22	eqeq1d 2740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐵 = 𝐷 ↔ 𝐶 = 𝐷))
24	23	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝐵 = 𝐷 ↔ 𝐶 = 𝐷)))
25	19, 21, 24	sbiedw 2313	. . . . . 6 ⊢ (𝜑 → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ 𝐶 = 𝐷))
26	18, 25	bitr3id 284	. . . . 5 ⊢ (𝜑 → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ 𝐶 = 𝐷))
27	26	pm5.74i 270	. . . 4 ⊢ ((𝜑 → [𝑦 / 𝑥]𝐵 = 𝐷) ↔ (𝜑 → 𝐶 = 𝐷))
28	13, 17, 27	vtoclbg 3497	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷) ↔ (𝜑 → 𝐷 = 𝐷)))
29	2, 28	mpbiri 257	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
30	1, 29	mpcom 38	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  Ⅎwnf 1787  [wsb 2068   ∈ wcel 2108  Ⅎwnfc 2886  [wsbc 3711  ⦋csb 3828

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

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-sbc 3712  df-csb 3829

This theorem is referenced by:  fvmptdf  6863