Theorem csbie2t 3153

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3154). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbie2t.1	⊢ 𝐴 ∈ V
csbie2t.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
csbie2t	⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷)

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

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem csbie2t

Step	Hyp	Ref	Expression
1		nfa1 1567	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
2		nfcvd 2353	. 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → Ⅎ𝑥𝐷)
3		csbie2t.1	. . 3 ⊢ 𝐴 ∈ V
4	3	a1i 9	. 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 ∈ V)
5		nfa2 1605	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
6		nfv 1554	. . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴
7	5, 6	nfan 1591	. . 3 ⊢ Ⅎ𝑦(∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴)
8		nfcvd 2353	. . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝐷)
9		csbie2t.2	. . . 4 ⊢ 𝐵 ∈ V
10	9	a1i 9	. . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11		sp 1537	. . . . 5 ⊢ (∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷))
12	11	sps 1563	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷))
13	12	impl 380	. . 3 ⊢ (((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
14	7, 8, 10, 13	csbiedf 3145	. 2 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌𝐶 = 𝐷)
15	1, 2, 4, 14	csbiedf 3145	1 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1373   = wceq 1375   ∈ wcel 2180  Vcvv 2779  ⦋csb 3104

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-sbc 3009  df-csb 3105

This theorem is referenced by:  csbie2  3154