Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ralcom3 | Structured version Visualization version GIF version |
Description: A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
ralcom3 | ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.04 90 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑)) → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑))) | |
2 | 1 | ralimi2 3084 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) → ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)) |
3 | pm2.04 90 | . . 3 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝜑))) | |
4 | 3 | ralimi2 3084 | . 2 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑)) |
5 | 2, 4 | impbii 208 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ∀wral 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ral 3069 |
This theorem is referenced by: tgss2 22137 ist1-3 22500 isreg2 22528 |
Copyright terms: Public domain | W3C validator |