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Mirrors > Home > MPE Home > Th. List > csbopabgALT | Structured version Visualization version GIF version |
Description: Move substitution into a class abstraction. Version of csbopab 5425 with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
csbopabgALT | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3805 | . . 3 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑}) | |
2 | dfsbcq2 3690 | . . . 4 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | opabbidv 5109 | . . 3 ⊢ (𝑤 = 𝐴 → {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
4 | 1, 3 | eqeq12d 2750 | . 2 ⊢ (𝑤 = 𝐴 → (⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} ↔ ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑})) |
5 | vex 3405 | . . 3 ⊢ 𝑤 ∈ V | |
6 | nfs1v 2157 | . . . 4 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑 | |
7 | 6 | nfopab 5111 | . . 3 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
8 | sbequ12 2249 | . . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) | |
9 | 8 | opabbidv 5109 | . . 3 ⊢ (𝑥 = 𝑤 → {〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑}) |
10 | 5, 7, 9 | csbief 3837 | . 2 ⊢ ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
11 | 4, 10 | vtoclg 3474 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 [wsb 2070 ∈ wcel 2110 [wsbc 3687 ⦋csb 3802 {copab 5105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-v 3403 df-sbc 3688 df-csb 3803 df-opab 5106 |
This theorem is referenced by: csbcnvgALT 5742 |
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