Theorem csbiebg 3887

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 2777	. . . 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 1943	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		csbeq1 3858	. . 3 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
5	4	eqeq1d 2767	. 2 ⊢ (𝑎 = 𝐴 → (⦋𝑎 / 𝑥⦌𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
6		vex 3461	. . 3 ⊢ 𝑎 ∈ V
7		csbiebg.2	. . 3 ⊢ Ⅎ𝑥𝐶
8	6, 7	csbieb 3886	. 2 ⊢ (∀𝑥(𝑥 = 𝑎 → 𝐵 = 𝐶) ↔ ⦋𝑎 / 𝑥⦌𝐵 = 𝐶)
9	3, 5, 8	vtoclbg 3527	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145  Ⅎwnfc 2912  ⦋csb 3855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-v 3459  df-sbc 3748  df-csb 3856

This theorem is referenced by:  cdlemefrs29bpre0  41032