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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsalhv | Structured version Visualization version GIF version |
Description: Version of equsalh 2413 with a disjoint variable condition, which
does not
require ax-13 2365. Remark: this is the same as equsalhw 2279. TODO:
delete after moving the following paragraph somewhere.
Remarks: equsexvw 2000 has been moved to Main; Theorem ax13lem2 2369 has a DV version which is a simple consequence of ax5e 1907; Theorems nfeqf2 2370, dveeq2 2371, nfeqf1 2372, dveeq1 2373, nfeqf 2374, axc9 2375, ax13 2368, have dv versions which are simple consequences of ax-5 1905. (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 2134 | . 2 ⊢ Ⅎ𝑥𝜓 |
3 | bj-equsalhv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | equsalv 2250 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-ex 1774 df-nf 1778 |
This theorem is referenced by: (None) |
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