![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > csbopabgALT | Structured version Visualization version GIF version |
Description: Move substitution into a class abstraction. Version of csbopab 5295 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 3791 | . . 3 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑}) | |
2 | dfsbcq2 3686 | . . . 4 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | opabbidv 4996 | . . 3 ⊢ (𝑤 = 𝐴 → {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
4 | 1, 3 | eqeq12d 2793 | . 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 3418 | . . 3 ⊢ 𝑤 ∈ V | |
6 | nfs1v 2202 | . . . 4 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑 | |
7 | 6 | nfopab 4998 | . . 3 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
8 | sbequ12 2179 | . . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) | |
9 | 8 | opabbidv 4996 | . . 3 ⊢ (𝑥 = 𝑤 → {〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑}) |
10 | 5, 7, 9 | csbief 3815 | . 2 ⊢ ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
11 | 4, 10 | vtoclg 3486 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 [wsb 2015 ∈ wcel 2050 [wsbc 3683 ⦋csb 3788 {copab 4992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-v 3417 df-sbc 3684 df-csb 3789 df-opab 4993 |
This theorem is referenced by: csbcnvgALT 5606 |
Copyright terms: Public domain | W3C validator |