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Mirrors > Home > ILE Home > Th. List > ralbid2 | GIF version |
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.) |
Ref | Expression |
---|---|
ralbid2.nf | ⊢ Ⅎ𝑥𝜑 |
ralbid2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) |
Ref | Expression |
---|---|
ralbid2 | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbid2.nf | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralbid2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) | |
3 | 1, 2 | albid 1592 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) |
4 | df-ral 2437 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
5 | df-ral 2437 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)) | |
6 | 3, 4, 5 | 3bitr4g 222 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 Ⅎwnf 1437 ∈ wcel 2125 ∀wral 2432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-4 1487 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-ral 2437 |
This theorem is referenced by: strcollnft 13497 |
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