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Mirrors > Home > MPE Home > Th. List > raleleqALT | Structured version Visualization version GIF version |
Description: Alternate proof of raleleq 3425 using ralel 3146, being longer and using more axioms. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
raleleqALT | ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralel 3146 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵 | |
2 | id 22 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
3 | 2 | raleqdv 3413 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐵)) |
4 | 1, 3 | mpbiri 259 | 1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 df-ral 3140 |
This theorem is referenced by: (None) |
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