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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dral1v | Structured version Visualization version GIF version |
Description: Version of dral1 2325 with a dv condition, which does not require ax-13 2246. Remark: the corresponding versions for dral2 2324 and drex2 2328 are instances of albidv 1849 and exbidv 1850 respectively. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dral1v.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-dral1v | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2028 | . . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
2 | bj-dral1v.1 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | albid 2090 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
4 | bj-axc11v 32747 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓)) | |
5 | axc11r 2187 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | impbid 202 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓)) |
7 | 3, 6 | bitrd 268 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: bj-drex1v 32749 bj-drnf1v 32750 |
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