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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 2465 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2377. This proof uses ax12 2428, see equs5aALT 2369 for an alternative one using ax-12 2177 but not ax13 2380. Usage of the weaker equs5av 2277 is preferred, which uses ax12v2 2179, but not ax-13 2377. (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 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑) | |
| 2 | ax12 2428 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
| 3 | 2 | imp 406 | . 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| 4 | 1, 3 | exlimi 2217 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: equs45f 2464 |
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