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Mirrors > Home > MPE Home > Th. List > equs5eALT | Structured version Visualization version GIF version |
Description: Alternate proof of equs5e 2458. Uses ax-12 2172 but not ax-13 2372. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equs5eALT | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2149 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) | |
2 | hbe1 2140 | . . . . 5 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑) | |
3 | 2 | 19.23bi 2185 | . . . 4 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑) |
4 | ax-12 2172 | . . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))) | |
5 | 3, 4 | syl5 34 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))) |
6 | 5 | imp 408 | . 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
7 | 1, 6 | exlimi 2211 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∃wex 1782 |
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-10 2138 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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