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Mirrors > Home > MPE Home > Th. List > ralanidOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralanid 3168 as of 29-Jun-2023. (Contributed by Peter Mazsa, 30-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ralanidOLD | ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anclb 548 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | 1 | albii 1820 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜑))) |
3 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
5 | 2, 3, 4 | 3bitr4ri 306 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∈ wcel 2114 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ral 3143 |
This theorem is referenced by: (None) |
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