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Mirrors > Home > MPE Home > Th. List > csbeq2d | Structured version Visualization version GIF version |
Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
csbeq2d.1 | ⊢ Ⅎ𝑥𝜑 |
csbeq2d.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbeq2d | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq2d.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | csbeq2d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | eleq2d 2868 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶)) |
4 | 1, 3 | sbcbid 3755 | . . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) |
5 | 4 | abbidv 2860 | . 2 ⊢ (𝜑 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
6 | df-csb 3812 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
7 | df-csb 3812 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
8 | 5, 6, 7 | 3eqtr4g 2856 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 Ⅎwnf 1765 ∈ wcel 2081 {cab 2775 [wsbc 3706 ⦋csb 3811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-sbc 3707 df-csb 3812 |
This theorem is referenced by: csbeq2dv 3818 poimirlem26 34449 |
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