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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 2869 | . 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 1364 ∈ wcel 2160 ∀wral 2468 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 |
This theorem is referenced by: swopo 4324 ordtri2orexmid 4540 onsucelsucexmid 4547 ordsucunielexmid 4548 ordtri2or2exmid 4588 ontri2orexmidim 4589 isocnv 5833 isotr 5838 ovrspc2v 5922 off 6119 caofrss 6131 oprssdmm 6196 tridc 6927 fidcenumlemrks 6982 seq3caopr2 10513 seq3distr 10544 isprm6 12179 mhmpropd 12918 grpidssd 13020 grpinvssd 13021 dfgrp3mlem 13042 isnsg3 13146 comet 14456 mulcncf 14548 trilpo 15250 neapmkv 15275 |
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