Theorem csbiedf 3880

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 416	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
4	1, 3	alrimi 2247	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
5		csbiedf.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		csbiedf.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶)
7		csbiebt 3879	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 6, 7	syl2anc 593	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	4, 8	mpbid 234	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  Ⅎwnf 1802   ∈ wcel 2141  Ⅎwnfc 2908  ⦋csb 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-sbc 3743  df-csb 3851

This theorem is referenced by:  csbie2t  3888  fvmptdf  6977  fsumsplit1  15763  fprodsplit1f  16011  natpropd  18003  fucpropd  18004  gsummptf1o  19994  gsummpt2d  33190  gsummptf1od  33196  gsummptfsf1o  33201  mnringvald  44750  sumsnd  45567