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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dral1v | Structured version Visualization version GIF version |
Description: Version of dral1 2313 with a dv condition, which does not require ax-13 2234. Remark: the corresponding versions for dral2 2312 and drex2 2316 are instances of albidv 1836 and exbidv 1837 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 2015 | . . 3 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
2 | bj-dral1v.1 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | albid 2077 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
4 | bj-axc11v 31935 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓)) | |
5 | axc11r 2175 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜓)) | |
6 | 4, 5 | impbid 201 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜓)) |
7 | 3, 6 | bitrd 267 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-drex1v 31937 bj-drnf1v 31938 |
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