Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > equs5e | Structured version Visualization version GIF version |
Description: A property related to substitution that unlike equs5 2339 does not require a distinctor antecedent. See equs5eALT 2166 for an alternate proof using ax-12 2034 but not ax13 2237. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) |
Ref | Expression |
---|---|
equs5e | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2015 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) | |
2 | ax12 2292 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))) | |
3 | hbe1 2008 | . . . 4 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑) | |
4 | 3 | 19.23bi 2049 | . . 3 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑) |
5 | 2, 4 | impel 484 | . 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
6 | 1, 5 | exlimi 2073 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: sb4e 2350 |
Copyright terms: Public domain | W3C validator |