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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 3624 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 3722 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | elrabsf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | nfsbc1v 3740 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
6 | sbceq1a 3731 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
7 | 2, 3, 4, 5, 6 | cbvrabw 3437 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑} |
8 | 1, 7 | elrab2 3631 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Ⅎwnfc 2936 {crab 3110 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-sbc 3721 |
This theorem is referenced by: wfisg 6151 onminesb 7493 mpoxopovel 7869 ac6num 9890 hashrabsn1 13731 bnj23 32098 bnj1204 32394 tfisg 33168 frpoinsg 33194 frinsg 33200 rabrenfdioph 39755 |
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