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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 2454 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2366. This proof uses ax12 2417, see equs5aALT 2358 for an alternative one using ax-12 2167 but not ax13 2369. Usage of the weaker equs5av 2266 is preferred, which uses ax12v2 2169, but not ax-13 2366. (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 2141 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
2 | ax12 2417 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 2 | imp 405 | . 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 1, 3 | exlimi 2206 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1532 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2366 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 df-nf 1779 |
This theorem is referenced by: equs45f 2453 |
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