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Mirrors > Home > ILE Home > Th. List > r19.21t | GIF version |
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.) |
Ref | Expression |
---|---|
r19.21t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi2.04 247 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))) | |
2 | 1 | albii 1463 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓))) |
3 | 19.21t 1575 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))) | |
4 | 2, 3 | syl5bb 191 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))) |
5 | df-ral 2453 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
6 | df-ral 2453 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
7 | 6 | imbi2i 225 | . 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
8 | 4, 5, 7 | 3bitr4g 222 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 Ⅎwnf 1453 ∈ wcel 2141 ∀wral 2448 |
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 1440 ax-gen 1442 ax-4 1503 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-ral 2453 |
This theorem is referenced by: r19.21 2546 |
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