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Mirrors > Home > MPE Home > Th. List > axc11r | Structured version Visualization version GIF version |
Description: Same as axc11 2429 but with reversed antecedent. Note the use
of ax-12 2171
(and not merely ax12v 2172 as in axc11rv 2257).
This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2056 and aecom 2426, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations. As of 2024, it is used in conjunction with ax-13 2371 only, and like that, it should be applied only in niches where indispensable. (Contributed by NM, 25-Jul-2015.) |
Ref | Expression |
---|---|
axc11r | ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-12 2171 | . . 3 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) | |
2 | 1 | sps 2178 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) |
3 | pm2.27 42 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑)) | |
4 | 3 | al2imi 1817 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑)) |
5 | 2, 4 | syld 47 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-ex 1782 |
This theorem is referenced by: ax12 2422 axc11n 2425 axc11 2429 hbae 2430 dral1 2438 dral1ALT 2439 sb4a 2483 axpowndlem3 10468 axc11n11r 35043 bj-ax12v3ALT 35046 bj-hbaeb2 35178 |
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