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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 5008 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 3536 | . . 3 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑}) | |
2 | dfsbcq2 3438 | . . . 4 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | opabbidv 4716 | . . 3 ⊢ (𝑤 = 𝐴 → {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
4 | 1, 3 | eqeq12d 2637 | . 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 3203 | . . 3 ⊢ 𝑤 ∈ V | |
6 | nfs1v 2437 | . . . 4 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑 | |
7 | 6 | nfopab 4718 | . . 3 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
8 | sbequ12 2111 | . . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) | |
9 | 8 | opabbidv 4716 | . . 3 ⊢ (𝑥 = 𝑤 → {〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑}) |
10 | 5, 7, 9 | csbief 3558 | . 2 ⊢ ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
11 | 4, 10 | vtoclg 3266 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 [wsb 1880 ∈ wcel 1990 [wsbc 3435 ⦋csb 3533 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 df-csb 3534 df-opab 4713 |
This theorem is referenced by: csbcnvgALT 5307 |
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