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| Mirrors > Home > MPE Home > Th. List > spc2ev | Structured version Visualization version GIF version | ||
| Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
| Ref | Expression |
|---|---|
| spc2ev.1 | ⊢ 𝐴 ∈ V |
| spc2ev.2 | ⊢ 𝐵 ∈ V |
| spc2ev.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| spc2ev | ⊢ (𝜓 → ∃𝑥∃𝑦𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spc2ev.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | spc2ev.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | spc2ev.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | spc2egv 3599 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑)) |
| 5 | 1, 2, 4 | mp2an 692 | 1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-clel 2816 |
| This theorem is referenced by: relop 5861 endisj 9098 dcomex 10487 axcnre 11204 hashle2pr 14516 wlk2f 29648 uhgr3cyclex 30201 qqhval2 33983 satfv1 35368 itg2addnclem3 37680 funop1 47295 |
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