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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbabgOLD | Structured version Visualization version GIF version |
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 3960 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
csbabgOLD | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbccom 3476 | . . . 4 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
2 | df-clab 2597 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
3 | sbsbc 3406 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
4 | 2, 3 | bitri 263 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) |
5 | df-clab 2597 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
6 | sbsbc 3406 | . . . . . 6 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
7 | 5, 6 | bitri 263 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) |
8 | 7 | sbcbii 3458 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) |
9 | 1, 4, 8 | 3bitr4i 291 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑}) |
10 | sbcel2gOLD 37776 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑} ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})) | |
11 | 9, 10 | syl5rbb 272 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑})) |
12 | 11 | eqrdv 2608 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 [wsb 1867 ∈ wcel 1977 {cab 2596 [wsbc 3402 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: csbunigOLD 38073 csbxpgOLD 38075 csbrngOLD 38078 csbingVD 38142 csbsngVD 38151 csbxpgVD 38152 csbrngVD 38154 csbunigVD 38156 |
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