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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 3623 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
4 | 3 | imp 408 | 1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 |
This theorem is referenced by: rspc2dv 3627 swopo 5600 soisores 7324 soisoi 7325 isocnv 7327 isotr 7333 ovrspc2v 7435 caofrss 7706 caonncan 7711 frpoins3xpg 8126 coflton 8670 wunpr 10704 injresinj 13753 seqcaopr2 14004 rlimcn3 15534 o1of2 15557 isprm6 16651 ssc2 17769 pospropd 18280 tleile 18374 mhmpropd 18678 grpidssd 18899 grpinvssd 18900 dfgrp3lem 18921 isnsg3 19040 cyccom 19080 symgextf1 19289 efgredlemd 19612 efgredlem 19615 rglcom4d 20034 issrngd 20469 domneq0 20913 lindfind 21371 lindsind 21372 mplsubglem 21558 mdetunilem1 22114 mdetunilem3 22116 mdetunilem4 22117 mdetunilem9 22122 decpmatmulsumfsupp 22275 pm2mpf1 22301 pm2mpmhmlem1 22320 t0sep 22828 tsmsxplem2 23658 comet 24022 nrmmetd 24083 tngngp 24171 reconnlem2 24343 iscmet3lem1 24808 iscmet3lem2 24809 dchrisumlem1 26992 pntpbnd1 27089 ssltsepc 27295 tgjustc1 27757 tgjustc2 27758 iscgrglt 27796 motcgr 27818 perpneq 27996 foot 28004 f1otrg 28153 axcontlem10 28262 frgr2wwlk1 29613 orngmul 32452 lindsunlem 32740 mndpluscn 32937 unelros 33200 difelros 33201 inelsros 33207 diffiunisros 33208 cvxsconn 34265 elmrsubrn 34542 ghomco 36807 sticksstones10 41019 sticksstones12a 41021 fsuppind 41210 mzpcl34 41517 ntrk0kbimka 42838 isotone1 42847 isotone2 42848 nnfoctbdjlem 45219 isomuspgrlem2d 46547 mgmhmpropd 46603 rnghmmul 46746 2arymaptf1 47387 |
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