Theorem csbieb 3126

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 3124	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
4	1, 2, 3	mp2an 426	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2167  Ⅎwnfc 2326  Vcvv 2763  ⦋csb 3084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990  df-csb 3085

This theorem is referenced by:  csbiebg  3127