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Mirrors > Home > MPE Home > Th. List > ralxfr | Structured version Visualization version GIF version |
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
Ref | Expression |
---|---|
ralxfr.1 | ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) |
ralxfr.2 | ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) |
ralxfr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralxfr | ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralxfr.1 | . . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) | |
2 | 1 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
3 | ralxfr.2 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
4 | 3 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) |
5 | ralxfr.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 5 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
7 | 2, 4, 6 | ralxfrd 5286 | . 2 ⊢ (⊤ → (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)) |
8 | 7 | mptru 1550 | 1 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ⊤wtru 1544 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 |
This theorem is referenced by: rexxfr 5294 infm3 11756 |
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