Theorem csbieb 3174

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 3172	. 2 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) ↔ [A / x]B = C))
4	1, 2, 3	mp2an 653	1 ⊢ (∀x(x = A → B = C) ↔ [A / x]B = C)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  csbiebg  3175