Theorem csbiebg 3861

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 2750	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑥 = 𝑎 ↔ 𝑥 = 𝐴))
2	1	imbi1d 341	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	albidv 1924	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		csbeq1 3831	. . 3 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
5	4	eqeq1d 2740	. 2 ⊢ (𝑎 = 𝐴 → (⦋𝑎 / 𝑥⦌𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6		vex 3426	. . 3 ⊢ 𝑎 ∈ V
7		csbiebg.2	. . 3 ⊢ Ⅎ𝑥𝐶
8	6, 7	csbieb 3860	. 2 ⊢ (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ⦋𝑎 / 𝑥⦌𝐵 = 𝐶)
9	3, 5, 8	vtoclbg 3497	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  ⦋csb 3828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424  df-sbc 3712  df-csb 3829

This theorem is referenced by:  cdlemefrs29bpre0  38337