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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vtoclgfALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of vtoclgf 3569. Proof from vtoclgft 3552. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-vtoclgfALT.1 | ⊢ Ⅎ𝑥𝐴 | 
| bj-vtoclgfALT.2 | ⊢ Ⅎ𝑥𝜓 | 
| bj-vtoclgfALT.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| bj-vtoclgfALT.4 | ⊢ 𝜑 | 
| Ref | Expression | 
|---|---|
| bj-vtoclgfALT | ⊢ (𝐴 ∈ 𝑉 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-vtoclgfALT.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 2 | bj-vtoclgfALT.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 1, 2 | pm3.2i 470 | . 2 ⊢ (Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) | 
| 4 | bj-vtoclgfALT.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 5 | 4 | ax-gen 1795 | . . 3 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| 6 | bj-vtoclgfALT.4 | . . . 4 ⊢ 𝜑 | |
| 7 | 6 | ax-gen 1795 | . . 3 ⊢ ∀𝑥𝜑 | 
| 8 | 5, 7 | pm3.2i 470 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) | 
| 9 | vtoclgft 3552 | . 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓) | |
| 10 | 3, 8, 9 | mp3an12 1453 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 | 
| This theorem is referenced by: (None) | 
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