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Mirrors > Home > NFE Home > Th. List > rspcev | GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) |
Ref | Expression |
---|---|
rspcv.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rspcev | ⊢ ((A ∈ B ∧ ψ) → ∃x ∈ B φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
2 | rspcv.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
3 | 1, 2 | rspce 2950 | 1 ⊢ ((A ∈ B ∧ ψ) → ∃x ∈ B φ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 |
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-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 |
This theorem is referenced by: rspc2ev 2963 rspc3ev 2965 reu6i 3027 rspesbca 3126 pwadjoin 4119 eqpw1uni 4330 nnc0suc 4412 elsuci 4414 0fin 4423 nnsucelr 4428 snfi 4431 lefinaddc 4450 nulge 4456 leltfintr 4458 ltfintr 4459 ltfinp1 4462 lefinlteq 4463 lefinrflx 4467 ltlefin 4468 ssfin 4470 ncfinraise 4481 ncfinlower 4483 tfinltfinlem1 4500 0ceven 4505 sucoddeven 4511 eventfin 4517 oddtfin 4518 nnpweq 4523 vfinspss 4551 phi11lem1 4595 foco2 5426 f1elima 5474 f1oiso2 5500 clos1conn 5879 clos1basesuc 5882 ecelqsg 5979 ncspw1eu 6159 pw1eltc 6162 nntccl 6170 lecidg 6196 lecncvg 6199 dflec2 6210 dflec3 6221 lenc 6223 letc 6231 ce0t 6232 ce0lenc1 6239 ncfin 6247 nmembers1lem3 6270 nncdiv3 6277 nchoicelem13 6301 nchoicelem17 6305 nchoicelem19 6307 nchoice 6308 frecsuc 6322 |
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