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Mirrors > Home > MPE Home > Th. List > elabgf | Structured version Visualization version GIF version |
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
elabgf.1 | ⊢ Ⅎ𝑥𝐴 |
elabgf.2 | ⊢ Ⅎ𝑥𝜓 |
elabgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elabgf | ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabgf.1 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfab1 2899 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | 1, 2 | nfel 2911 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
4 | elabgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfbi 1911 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) |
6 | eleq1 2818 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
7 | elabgf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | bibi12d 349 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))) |
9 | abid 2718 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
10 | 1, 5, 8, 9 | vtoclgf 3469 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2112 {cab 2714 Ⅎwnfc 2877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-v 3400 |
This theorem is referenced by: elabf 3573 elab3gf 3582 elrabf 3587 currysetlem 34820 currysetlem1 34822 |
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