Theorem csbie2t 3876

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3877). (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 2162	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
2		nfcvd 2903	. 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → Ⅎ𝑥𝐷)
3		csbie2t.1	. . 3 ⊢ 𝐴 ∈ V
4	3	a1i 11	. 2 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 ∈ V)
5		nfa2 2186	. . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
6		nfv 1921	. . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴
7	5, 6	nfan 1906	. . 3 ⊢ Ⅎ𝑦(∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴)
8		nfcvd 2903	. . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝐷)
9		csbie2t.2	. . . 4 ⊢ 𝐵 ∈ V
10	9	a1i 11	. . 3 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11		2sp 2198	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷))
12	11	impl 456	. . 3 ⊢ (((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷)
13	7, 8, 10, 12	csbiedf 3868	. 2 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) ∧ 𝑥 = 𝐴) → ⦋𝐵 / 𝑦⦌𝐶 = 𝐷)
14	1, 2, 4, 13	csbiedf 3868	1 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ⦋csb 3838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-v 3434  df-sbc 3731  df-csb 3839

This theorem is referenced by:  csbie2  3877