Theorem csbiedf 3863

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 413	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
4	1, 3	alrimi 2206	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
5		csbiedf.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		csbiedf.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶)
7		csbiebt 3862	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 6, 7	syl2anc 584	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	4, 8	mpbid 231	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ⦋csb 3832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-sbc 3717  df-csb 3833

This theorem is referenced by:  csbiedOLD  3871  csbie2t  3873  fvmptdf  6881  fsumsplit1  15457  fprodsplit1f  15700  natpropd  17694  fucpropd  17695  gsummptf1o  19564  gsummpt2d  31309  mnringvald  41826  sumsnd  42569