Theorem csbie2g 3000

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2895 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Hypotheses

Ref	Expression
csbie2g.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
csbie2g.2	⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)

Assertion

Ref	Expression
csbie2g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑦,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbie2g

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 2956	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
2		csbie2g.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	2	eleq2d 2169	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
4		csbie2g.2	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)
5	4	eleq2d 2169	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
6	3, 5	sbcie2g 2894	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐷))
7	6	abbi1dv 2219	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = 𝐷)
8	1, 7	syl5eq 2144	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  {cab 2086  [wsbc 2862  ⦋csb 2955

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863  df-csb 2956

This theorem is referenced by: (None)