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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 3622 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
4 | 3 | imp 407 | 1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 |
This theorem is referenced by: rspc2dv 3626 swopo 5599 soisores 7323 soisoi 7324 isocnv 7326 isotr 7332 ovrspc2v 7434 caofrss 7705 caonncan 7710 frpoins3xpg 8125 coflton 8669 wunpr 10703 injresinj 13752 seqcaopr2 14003 rlimcn3 15533 o1of2 15556 isprm6 16650 ssc2 17768 pospropd 18279 tleile 18373 mhmpropd 18677 grpidssd 18898 grpinvssd 18899 dfgrp3lem 18920 isnsg3 19039 cyccom 19079 symgextf1 19288 efgredlemd 19611 efgredlem 19614 rglcom4d 20033 issrngd 20468 domneq0 20912 lindfind 21370 lindsind 21371 mplsubglem 21557 mdetunilem1 22113 mdetunilem3 22115 mdetunilem4 22116 mdetunilem9 22121 decpmatmulsumfsupp 22274 pm2mpf1 22300 pm2mpmhmlem1 22319 t0sep 22827 tsmsxplem2 23657 comet 24021 nrmmetd 24082 tngngp 24170 reconnlem2 24342 iscmet3lem1 24807 iscmet3lem2 24808 dchrisumlem1 26989 pntpbnd1 27086 ssltsepc 27291 tgjustc1 27723 tgjustc2 27724 iscgrglt 27762 motcgr 27784 perpneq 27962 foot 27970 f1otrg 28119 axcontlem10 28228 frgr2wwlk1 29579 orngmul 32416 lindsunlem 32704 mndpluscn 32901 unelros 33164 difelros 33165 inelsros 33171 diffiunisros 33172 cvxsconn 34229 elmrsubrn 34506 ghomco 36754 sticksstones10 40966 sticksstones12a 40968 fsuppind 41164 mzpcl34 41459 ntrk0kbimka 42780 isotone1 42789 isotone2 42790 nnfoctbdjlem 45161 isomuspgrlem2d 46489 mgmhmpropd 46545 rnghmmul 46688 2arymaptf1 47329 |
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