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Mirrors > Home > MPE Home > Th. List > rspe | Structured version Visualization version GIF version |
Description: Restricted specialization. (Contributed by NM, 12-Oct-1999.) |
Ref | Expression |
---|---|
rspe | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2039 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | 1, 2 | sylibr 223 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-rex 2902 |
This theorem is referenced by: rsp2e 2987 2rmorex 3379 ssiun2 4499 reusv2lem3 4797 tfrlem9 7368 isinf 8058 findcard2 8085 findcard3 8088 scott0 8632 ac6c4 9186 supaddc 10867 supadd 10868 supmul1 10869 supmul 10872 fsuppmapnn0fiub 12652 mreiincl 16079 restmetu 22185 bposlem3 24811 opphllem5 25443 pjpjpre 27662 atom1d 28596 bnj1398 30356 cvmlift2lem12 30550 finminlem 31482 neibastop2lem 31525 iooelexlt 32386 relowlpssretop 32388 prtlem18 33180 pell14qrdich 36451 eliuniin 38307 eliuniin2 38335 disjinfi 38375 iunmapsn 38404 upbdrech 38460 limclner 38718 sge0iunmptlemre 39308 iundjiun 39353 meaiininclem 39376 isomenndlem 39420 ovnsubaddlem1 39460 vonioo 39573 vonicc 39576 smfaddlem1 39649 2reurex 39830 |
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