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Mirrors > Home > MPE Home > Th. List > equs5e | Structured version Visualization version GIF version |
Description: A property related to substitution that unlike equs5 2457 does not require a distinctor antecedent. This proof uses ax12 2420, see equs5eALT 2362 for an alternative one using ax-12 2169 but not ax13 2372. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equs5e | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2146 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) | |
2 | ax12 2420 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))) | |
3 | hbe1 2137 | . . . 4 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑) | |
4 | 3 | 19.23bi 2182 | . . 3 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑) |
5 | 2, 4 | impel 504 | . 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
6 | 1, 5 | exlimi 2208 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1537 ∃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 1911 ax-6 1969 ax-7 2009 ax-10 2135 ax-12 2169 ax-13 2369 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: sb4e 2482 |
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