Theorem csbie2 3921

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 1793	. 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 3920	. 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 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ⦋csb 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3772  df-csb 3883

This theorem is referenced by:  fsumcnv  15127  fprodcnv  15336  dfrhm2  19468  mamufval  20995  mvmulfval  21150  vtxdgfval  27248  rnghmval  44161