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Mirrors > Home > MPE Home > Th. List > csbie2g | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3921 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
csbie2g.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
csbie2g.2 | ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
csbie2g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3887 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} | |
2 | csbie2g.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
3 | 2 | eleq2d 2901 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
4 | csbie2g.2 | . . . . 5 ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐷) | |
5 | 4 | eleq2d 2901 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
6 | 3, 5 | sbcie2g 3814 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐷)) |
7 | 6 | abbi1dv 2955 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} = 𝐷) |
8 | 1, 7 | syl5eq 2871 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {cab 2802 [wsbc 3775 ⦋csb 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-sbc 3776 df-csb 3887 |
This theorem is referenced by: (None) |
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