Theorem csbie2df 4348

Description: Conversion of implicit substitution to explicit class substitution. This version of csbiedf 3858 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. Deduction form of csbie2 3867. (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 2799	. . 3 ⊢ (𝜑 → 𝐷 = 𝐷)
3		dfsbcq 3722	. . . . . 6 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ [𝐴 / 𝑥]𝐵 = 𝐷))
4		sbceqg 4317	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐷))
5	4	adantr 484	. . . . . . . 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 595	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ⦋𝐴 / 𝑥⦌𝐷 = 𝐷)
9	8	eqeq2d 2809	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
10	5, 9	bitrd 282	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
11	1, 10	mpancom 687	. . . . . 6 ⊢ (𝜑 → ([𝐴 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
12	3, 11	sylan9bb 513	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
13	12	pm5.74da 803	. . . 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 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐷 = 𝐷))
16	15	expcom 417	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝜑 → (𝐶 = 𝐷 ↔ 𝐷 = 𝐷)))
17	16	pm5.74d 276	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝐶 = 𝐷) ↔ (𝜑 → 𝐷 = 𝐷)))
18		sbsbc 3724	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ [𝑦 / 𝑥]𝐵 = 𝐷)
19		csbie2df.p	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
20		csbie2df.c	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐶)
21	20, 6	nfeqd 2965	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝐶 = 𝐷)
22		csbie2df.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
23	22	eqeq1d 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝐵 = 𝐷 ↔ 𝐶 = 𝐷))
24	23	ex 416	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝐵 = 𝐷 ↔ 𝐶 = 𝐷)))
25	19, 21, 24	sbiedw 2323	. . . . . 6 ⊢ (𝜑 → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ 𝐶 = 𝐷))
26	18, 25	bitr3id 288	. . . . 5 ⊢ (𝜑 → ([𝑦 / 𝑥]𝐵 = 𝐷 ↔ 𝐶 = 𝐷))
27	26	pm5.74i 274	. . . 4 ⊢ ((𝜑 → [𝑦 / 𝑥]𝐵 = 𝐷) ↔ (𝜑 → 𝐶 = 𝐷))
28	13, 17, 27	vtoclbg 3517	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷) ↔ (𝜑 → 𝐷 = 𝐷)))
29	2, 28	mpbiri 261	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷))
30	1, 29	mpcom 38	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  Ⅎwnfc 2936  [wsbc 3720  ⦋csb 3828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-sbc 3721  df-csb 3829

This theorem is referenced by:  fvmptdf  6751