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Mirrors > Home > MPE Home > Th. List > vtoclegftOLD | Structured version Visualization version GIF version |
Description: Obsolete version of vtoclegft 3568 as of 26-Jan-2025. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vtoclegftOLD | ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2809 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
2 | exim 1828 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑)) | |
3 | 1, 2 | mpan9 506 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → ∃𝑥𝜑) |
4 | 3 | 3adant2 1128 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → ∃𝑥𝜑) |
5 | 19.9t 2189 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
6 | 5 | 3ad2ant2 1131 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → (∃𝑥𝜑 ↔ 𝜑)) |
7 | 4, 6 | mpbid 231 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∀wal 1531 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 ∈ wcel 2098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-clel 2804 |
This theorem is referenced by: (None) |
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