Theorem csbie2g 2896

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2797 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 2853	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
2		csbie2g.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	2	eleq2d 2107	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
4		csbie2g.2	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)
5	4	eleq2d 2107	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
6	3, 5	sbcie2g 2796	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐷))
7	6	abbi1dv 2157	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = 𝐷)
8	1, 7	syl5eq 2084	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  {cab 2026  [wsbc 2764  ⦋csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by: (None)