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Mirrors > Home > MPE Home > Th. List > sbcbid | Structured version Visualization version GIF version |
Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) |
Ref | Expression |
---|---|
sbcbid.1 | ⊢ Ⅎ𝑥𝜑 |
sbcbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbcbid | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | sbcbid.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | abbid 2884 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
4 | 3 | eleq2d 2895 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜒})) |
5 | df-sbc 3770 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
6 | df-sbc 3770 | . 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝐴 ∈ {𝑥 ∣ 𝜒}) | |
7 | 4, 5, 6 | 3bitr4g 315 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1775 ∈ wcel 2105 {cab 2796 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-sbc 3770 |
This theorem is referenced by: sbcbidvOLD 3825 sbcbi2 3828 csbeq2d 3886 |
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