Theorem csbie2 3891

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 1824	. 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 3890	. 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 400  ∀wal 1566   = wceq 1568   ∈ wcel 2141  Vcvv 3453  ⦋csb 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-sbc 3744  df-csb 3853

This theorem is referenced by:  fsumcnv  15824  fprodcnv  16037  rnghmval  20521  dfrhm2  20555  mamufval  22528  mvmulfval  22678  vtxdgfval  29783  grtri  48672