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Mirrors > Home > MPE Home > Th. List > elabf | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
elabf.1 | ⊢ Ⅎ𝑥𝜓 |
elabf.2 | ⊢ 𝐴 ∈ V |
elabf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elabf | ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabf.2 | . 2 ⊢ 𝐴 ∈ V | |
2 | nfcv 2907 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | elabf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | elabf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 2, 3, 4 | elabgf 3606 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 {cab 2715 Vcvv 3431 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-v 3433 |
This theorem is referenced by: elabOLD 3611 dfon2lem1 33746 sdclem2 35887 sdclem1 35888 scottabf 41818 |
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