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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 3598 | . 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 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-clel 2813 |
This theorem is referenced by: relop 5863 endisj 9096 dcomex 10484 axcnre 11201 hashle2pr 14512 wlk2f 29662 uhgr3cyclex 30210 qqhval2 33944 satfv1 35347 itg2addnclem3 37659 funop1 47232 |
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