Theorem csbie2 3182

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Hypotheses

Ref	Expression
csbie2t.1	⊢ A ∈ V
csbie2t.2	⊢ B ∈ V
csbie2.3	⊢ ((x = A ∧ y = B) → C = D)

Assertion

Ref	Expression
csbie2	⊢ [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 csbie2

Step	Hyp	Ref	Expression
1		csbie2.3	. . 3 ⊢ ((x = A ∧ y = B) → C = D)
2	1	gen2 1547	. 2 ⊢ ∀x∀y((x = A ∧ y = B) → C = D)
3		csbie2t.1	. . 3 ⊢ A ∈ V
4		csbie2t.2	. . 3 ⊢ B ∈ V
5	3, 4	csbie2t 3181	. 2 ⊢ (∀x∀y((x = A ∧ y = B) → C = D) → [A / x][B / y]C = D)
6	2, 5	ax-mp 5	1 ⊢ [A / x][B / y]C = D

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)