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Mirrors > Home > NFE Home > Th. List > rexsn | GIF version |
Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Ref | Expression |
---|---|
ralsn.1 | ⊢ A ∈ V |
ralsn.2 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rexsn | ⊢ (∃x ∈ {A}φ ↔ ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsn.1 | . 2 ⊢ A ∈ V | |
2 | ralsn.2 | . . 3 ⊢ (x = A → (φ ↔ ψ)) | |
3 | 2 | rexsng 3766 | . 2 ⊢ (A ∈ V → (∃x ∈ {A}φ ↔ ψ)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (∃x ∈ {A}φ ↔ ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 Vcvv 2859 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-v 2861 df-sbc 3047 df-sn 3741 |
This theorem is referenced by: pw1sn 4165 elimaksn 4283 setswith 4321 addcid1 4405 ltfintrilem1 4465 nnadjoin 4520 tfinnn 4534 elsnres 4996 xpnedisj 5513 snec 5987 lec0cg 6198 addccan2nclem1 6263 |
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