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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax89 | Structured version Visualization version GIF version |
Description: A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2109 and ax-9 2117. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2109 and ax-9 2117, as proved here. In the other direction, one can prove ax-8 2109 (respectively ax-9 2117) from bj-ax89 35171 by using mpan2 690 (respectively mpan 689) and equid 2016. TODO: move to main part. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bj-ax89 | ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax8 2113 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
2 | ax9 2121 | . 2 ⊢ (𝑧 = 𝑡 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑡)) | |
3 | 1, 2 | sylan9 509 | 1 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: (None) |
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