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Mirrors > Home > ILE Home > Th. List > ralxfrALT | GIF version |
Description: Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4160. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralxfr.1 | ⊢ (y ∈ 𝐶 → A ∈ B) |
ralxfr.2 | ⊢ (x ∈ B → ∃y ∈ 𝐶 x = A) |
ralxfr.3 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
ralxfrALT | ⊢ (∀x ∈ B φ ↔ ∀y ∈ 𝐶 ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralxfr.1 | . . . . 5 ⊢ (y ∈ 𝐶 → A ∈ B) | |
2 | ralxfr.3 | . . . . . 6 ⊢ (x = A → (φ ↔ ψ)) | |
3 | 2 | rspcv 2646 | . . . . 5 ⊢ (A ∈ B → (∀x ∈ B φ → ψ)) |
4 | 1, 3 | syl 14 | . . . 4 ⊢ (y ∈ 𝐶 → (∀x ∈ B φ → ψ)) |
5 | 4 | com12 27 | . . 3 ⊢ (∀x ∈ B φ → (y ∈ 𝐶 → ψ)) |
6 | 5 | ralrimiv 2385 | . 2 ⊢ (∀x ∈ B φ → ∀y ∈ 𝐶 ψ) |
7 | ralxfr.2 | . . . 4 ⊢ (x ∈ B → ∃y ∈ 𝐶 x = A) | |
8 | nfra1 2349 | . . . . 5 ⊢ Ⅎy∀y ∈ 𝐶 ψ | |
9 | nfv 1418 | . . . . 5 ⊢ Ⅎyφ | |
10 | rsp 2363 | . . . . . 6 ⊢ (∀y ∈ 𝐶 ψ → (y ∈ 𝐶 → ψ)) | |
11 | 2 | biimprcd 149 | . . . . . 6 ⊢ (ψ → (x = A → φ)) |
12 | 10, 11 | syl6 29 | . . . . 5 ⊢ (∀y ∈ 𝐶 ψ → (y ∈ 𝐶 → (x = A → φ))) |
13 | 8, 9, 12 | rexlimd 2424 | . . . 4 ⊢ (∀y ∈ 𝐶 ψ → (∃y ∈ 𝐶 x = A → φ)) |
14 | 7, 13 | syl5 28 | . . 3 ⊢ (∀y ∈ 𝐶 ψ → (x ∈ B → φ)) |
15 | 14 | ralrimiv 2385 | . 2 ⊢ (∀y ∈ 𝐶 ψ → ∀x ∈ B φ) |
16 | 6, 15 | impbii 117 | 1 ⊢ (∀x ∈ B φ ↔ ∀y ∈ 𝐶 ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ∃wrex 2301 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 |
This theorem is referenced by: (None) |
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