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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 3680 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 3780 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | elrabsf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2904 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1918 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | nfsbc1v 3798 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
6 | sbceq1a 3789 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
7 | 2, 3, 4, 5, 6 | cbvrabw 3468 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑} |
8 | 1, 7 | elrab2 3687 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Ⅎwnfc 2884 {crab 3433 [wsbc 3778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-sbc 3779 |
This theorem is referenced by: frpoinsg 6345 wfisgOLD 6356 onminesb 7781 tfisg 7843 mpoxopovel 8205 frinsg 9746 ac6num 10474 hashrabsn1 14334 bnj23 33729 bnj1204 34023 rabrenfdioph 41552 |
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