![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elrabsf | Structured version Visualization version GIF version |
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3392 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
elrabsf.1 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
elrabsf | ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3470 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | elrabsf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2793 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1883 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | nfsbc1v 3488 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
6 | sbceq1a 3479 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
7 | 2, 3, 4, 5, 6 | cbvrab 3229 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑} |
8 | 1, 7 | elrab2 3399 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∈ wcel 2030 Ⅎwnfc 2780 {crab 2945 [wsbc 3468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-sbc 3469 |
This theorem is referenced by: wfisg 5753 onminesb 7040 mpt2xopovel 7391 ac6num 9339 hashrabsn1 13201 bnj23 30912 bnj1204 31206 tfisg 31840 frpoinsg 31866 frinsg 31870 rabrenfdioph 37695 |
Copyright terms: Public domain | W3C validator |