Theorem csbiedf 3174

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbiedf.1	⊢ Ⅎxφ
csbiedf.2	⊢ (φ → ℲxC)
csbiedf.3	⊢ (φ → A ∈ V)
csbiedf.4	⊢ ((φ ∧ x = A) → B = C)

Assertion

Ref	Expression
csbiedf	⊢ (φ → [A / x]B = C)

Distinct variable group:   x,A

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

Proof of Theorem csbiedf

Step	Hyp	Ref	Expression
1		csbiedf.1	. . 3 ⊢ Ⅎxφ
2		csbiedf.4	. . . 4 ⊢ ((φ ∧ x = A) → B = C)
3	2	ex 423	. . 3 ⊢ (φ → (x = A → B = C))
4	1, 3	alrimi 1765	. 2 ⊢ (φ → ∀x(x = A → B = C))
5		csbiedf.3	. . 3 ⊢ (φ → A ∈ V)
6		csbiedf.2	. . 3 ⊢ (φ → ℲxC)
7		csbiebt 3173	. . 3 ⊢ ((A ∈ V ∧ ℲxC) → (∀x(x = A → B = C) ↔ [A / x]B = C))
8	5, 6, 7	syl2anc 642	. 2 ⊢ (φ → (∀x(x = A → B = C) ↔ [A / x]B = C))
9	4, 8	mpbid 201	1 ⊢ (φ → [A / x]B = C)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  [csb 3137

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 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbied  3179  csbie2t  3181