Theorem csbieb 3885

Description: Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Hypotheses

Ref	Expression
csbieb.1	⊢ 𝐴 ∈ V
csbieb.2	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
csbieb	⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbieb

Step	Hyp	Ref	Expression
1		csbieb.1	. 2 ⊢ 𝐴 ∈ V
2		csbieb.2	. 2 ⊢ Ⅎ𝑥𝐶
3		csbiebt 3883	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
4	1, 2, 3	mp2an 702	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560   = wceq 1562   ∈ wcel 2144  Ⅎwnfc 2911  Vcvv 3456  ⦋csb 3854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-v 3458  df-sbc 3747  df-csb 3855

This theorem is referenced by:  csbiebg  3886