Theorem csbie2t 3180

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3181). (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 1788	. 2 ⊢ Ⅎx∀x∀y((x = A ∧ y = B) → C = D)
2		nfcvd 2490	. 2 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → ℲxD)
3		csbie2t.1	. . 3 ⊢ A ∈ V
4	3	a1i 10	. 2 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → A ∈ V)
5		nfa2 1855	. . . 4 ⊢ Ⅎy∀x∀y((x = A ∧ y = B) → C = D)
6		nfv 1619	. . . 4 ⊢ Ⅎy x = A
7	5, 6	nfan 1824	. . 3 ⊢ Ⅎy(∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A)
8		nfcvd 2490	. . 3 ⊢ ((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) → ℲyD)
9		csbie2t.2	. . . 4 ⊢ B ∈ V
10	9	a1i 10	. . 3 ⊢ ((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) → B ∈ V)
11		sp 1747	. . . . 5 ⊢ (∀y((x = A ∧ y = B) → C = D) → ((x = A ∧ y = B) → C = D))
12	11	sps 1754	. . . 4 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → ((x = A ∧ y = B) → C = D))
13	12	impl 603	. . 3 ⊢ (((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) ∧ y = B) → C = D)
14	7, 8, 10, 13	csbiedf 3173	. 2 ⊢ ((∀x∀y((x = A ∧ y = B) → C = D) ∧ x = A) → [B / y]C = D)
15	1, 2, 4, 14	csbiedf 3173	1 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → [A / x][B / y]C = D)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [csb 3136

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbie2  3181