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Mirrors > Home > MPE Home > Th. List > vtoclegft | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3254.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Ref | Expression |
---|---|
vtoclegft | ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 3188 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
2 | exim 1751 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑)) | |
3 | 1, 2 | mpan9 485 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → ∃𝑥𝜑) |
4 | 3 | 3adant2 1073 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → ∃𝑥𝜑) |
5 | 19.9t 2059 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
6 | 5 | 3ad2ant2 1076 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → (∃𝑥𝜑 ↔ 𝜑)) |
7 | 4, 6 | mpbid 221 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: vtoclefex 32357 |
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