Theorem csbie2 3886

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 1797	. 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 3885	. 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 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⦋csb 3847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-v 3440  df-sbc 3739  df-csb 3848

This theorem is referenced by:  fsumcnv  15690  fprodcnv  15900  rnghmval  20368  dfrhm2  20402  mamufval  22317  mvmulfval  22467  vtxdgfval  29457  grtri  48054