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Mirrors > Home > MPE Home > Th. List > ralanid | Structured version Visualization version GIF version |
Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018.) (Proof shortened by Wolf Lammen, 29-Jun-2023.) |
Ref | Expression |
---|---|
ralanid | ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 532 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | 1 | bicomd 226 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝜑)) |
3 | 2 | ralbiia 3132 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ral 3111 |
This theorem is referenced by: idinxpssinxp2 35735 |
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