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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsalhv | Structured version Visualization version GIF version |
Description: Version of equsalh 2280 with a dv condition, which does not require
ax-13 2234. Remark: this is the same as equsalhw 2109.
Remarks: equsexvw 1919 has been moved to Main; the theorem ax13lem2 2284 has a dv version which is a simple consequence of ax5e 1829; the theorems nfeqf2 2285, dveeq2 2286, nfeqf1 2287, dveeq1 2288, nfeqf 2289, axc9 2290, ax13 2237, have dv versions which are simple consequences of ax-5 1827. (Contributed by BJ, 14-Jun-2019.) (Proof modification 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 2011 | . 2 ⊢ Ⅎ𝑥𝜓 |
3 | bj-equsalhv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | bj-equsalv 31931 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
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-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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