Theorem csbiebg 3893

Description: Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴 → 𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
csbiebg.2	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
csbiebg	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem csbiebg

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2781	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑥 = 𝑎 ↔ 𝑥 = 𝐴))
2	1	imbi1d 344	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	albidv 1947	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		csbeq1 3864	. . 3 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
5	4	eqeq1d 2771	. 2 ⊢ (𝑎 = 𝐴 → (⦋𝑎 / 𝑥⦌𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6		vex 3467	. . 3 ⊢ 𝑎 ∈ V
7		csbiebg.2	. . 3 ⊢ Ⅎ𝑥𝐶
8	6, 7	csbieb 3892	. 2 ⊢ (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ⦋𝑎 / 𝑥⦌𝐵 = 𝐶)
9	3, 5, 8	vtoclbg 3533	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567   ∈ wcel 2149  Ⅎwnfc 2916  ⦋csb 3861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-v 3465  df-sbc 3754  df-csb 3862

This theorem is referenced by:  cdlemefrs29bpre0  41094