Theorem csbie2g 3896

Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3891 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 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 3857	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
2		csbie2g.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	2	eleq2d 2852	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
4		csbie2g.2	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)
5	4	eleq2d 2852	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
6	3, 5	sbcie2g 3787	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐷))
7	6	eqabcdv 2900	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = 𝐷)
8	1, 7	eqtrid 2813	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {cab 2744  [wsbc 3747  ⦋csb 3856

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-sbc 3748  df-csb 3857

This theorem is used by: (None)