Theorem csbie2g 3937

Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3930 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 3895	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
2		csbie2g.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	2	eleq2d 2815	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
4		csbie2g.2	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷)
5	4	eleq2d 2815	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
6	3, 5	sbcie2g 3821	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐷))
7	6	eqabcdv 2864	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = 𝐷)
8	1, 7	eqtrid 2780	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {cab 2705  [wsbc 3778  ⦋csb 3894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-sbc 3779  df-csb 3895

This theorem is referenced by: (None)