Theorem csbiedf 3920

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 411	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
4	1, 3	alrimi 2201	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
5		csbiedf.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		csbiedf.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐶)
7		csbiebt 3919	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 6, 7	syl2anc 582	. 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 394  ∀wal 1531   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875  ⦋csb 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-v 3463  df-sbc 3774  df-csb 3890

This theorem is referenced by:  csbiedOLD  3928  csbie2t  3930  fvmptdf  7010  fsumsplit1  15727  fprodsplit1f  15970  natpropd  17971  fucpropd  17972  gsummptf1o  19930  gsummpt2d  32853  mnringvald  43787  sumsnd  44530