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Mirrors > Home > ILE Home > Th. List > rspc2va | 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 2854 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
4 | 3 | imp 124 | 1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑) → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 |
This theorem is referenced by: swopo 4302 ordtri2orexmid 4518 onsucelsucexmid 4525 ordsucunielexmid 4526 ordtri2or2exmid 4566 ontri2orexmidim 4567 isocnv 5805 isotr 5810 ovrspc2v 5894 off 6088 caofrss 6100 oprssdmm 6165 tridc 6892 fidcenumlemrks 6945 seq3caopr2 10455 seq3distr 10486 isprm6 12117 mhmpropd 12734 grpidssd 12822 grpinvssd 12823 dfgrp3mlem 12844 comet 13632 mulcncf 13724 trilpo 14414 neapmkv 14438 |
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