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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsalhv | Structured version Visualization version GIF version |
Description: Version of equsalh 2436 with a disjoint variable condition, which
does not
require ax-13 2383. Remark: this is the same as equsalhw 2291. TODO:
delete after moving the following paragraph somewhere.
Remarks: equsexvw 2002 has been moved to Main; the theorem ax13lem2 2387 has a dv version which is a simple consequence of ax5e 1904; the theorems nfeqf2 2388, dveeq2 2389, nfeqf1 2390, dveeq1 2391, nfeqf 2392, axc9 2393, ax13 2386, have dv versions which are simple consequences of ax-5 1902. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-equsalhv.nf | ⊢ (𝜓 → ∀𝑥𝜓) |
bj-equsalhv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-equsalhv | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsalhv.nf | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | nf5i 2141 | . 2 ⊢ Ⅎ𝑥𝜓 |
3 | bj-equsalhv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | equsalv 2259 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 |
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-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: (None) |
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