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Mirrors > Home > MPE Home > Th. List > rspc2va | Structured version Visualization version GIF version |
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.) |
Ref | Expression |
---|---|
rspc2v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc2v.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc2va | ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc2v.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
2 | rspc2v.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) | |
3 | 1, 2 | rspc2v 3293 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
4 | 3 | imp 444 | 1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 |
This theorem is referenced by: swopo 4969 soisores 6477 soisoi 6478 isocnv 6480 isotr 6486 ovrspc2v 6571 off 6810 caofrss 6828 caonncan 6833 wunpr 9410 injresinj 12451 seqcaopr2 12699 rlimcn2 14169 o1of2 14191 isprm6 15264 ssc2 16305 pospropd 16957 mhmpropd 17164 grpidssd 17314 grpinvssd 17315 dfgrp3lem 17336 isnsg3 17451 symgextf1 17664 efgredlemd 17980 efgredlem 17983 issrngd 18684 domneq0 19118 mplsubglem 19255 lindfind 19974 lindsind 19975 mdetunilem1 20237 mdetunilem3 20239 mdetunilem4 20240 mdetunilem9 20245 decpmatmulsumfsupp 20397 pm2mpf1 20423 pm2mpmhmlem1 20442 t0sep 20938 tsmsxplem2 21767 comet 22128 nrmmetd 22189 tngngp 22268 reconnlem2 22438 iscmet3lem1 22897 iscmet3lem2 22898 dchrisumlem1 24978 pntpbnd1 25075 motcgr 25231 perpneq 25409 foot 25414 f1otrg 25551 axcontlem10 25653 frg2wot1 26584 tleile 28992 orngmul 29134 mndpluscn 29300 unelros 29561 difelros 29562 inelsros 29568 diffiunisros 29569 cvxscon 30479 elmrsubrn 30671 ghomco 32860 mzpcl34 36312 ntrk0kbimka 37357 isotone1 37366 isotone2 37367 nnfoctbdjlem 39348 frgr2wwlk1 41494 mgmhmpropd 41575 rnghmmul 41690 |
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