Theorem csbiedf 3554

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 450	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
4	1, 3	alrimi 2082	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
5		csbiedf.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		csbiedf.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶)
7		csbiebt 3553	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 6, 7	syl2anc 693	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	4, 8	mpbid 222	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  ⦋csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbied  3560  csbie2t  3562  fprodsplit1f  14721  natpropd  16636  fucpropd  16637  gsummptf1o  18362  gsummpt2d  29781  sumsnd  39185  fsumsplit1  39804