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Mirrors > Home > MPE Home > Th. List > equs5a | Structured version Visualization version GIF version |
Description: A property related to substitution that unlike equs5 2463 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2375. This proof uses ax12 2426, see equs5aALT 2367 for an alternative one using ax-12 2175 but not ax13 2378. Usage of the weaker equs5av 2275 is preferred, which uses ax12v2 2177, but not ax-13 2375. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equs5a | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2149 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
2 | ax12 2426 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 2 | imp 406 | . 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 1, 3 | exlimi 2215 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: equs45f 2462 |
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