![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equs4 | Structured version Visualization version GIF version |
Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sb56 2274) or a non-freeness hypothesis (equs45f 2471). Usage of this theorem is discouraged because it depends on ax-13 2379. See equs4v 2006 for a weaker version requiring fewer axioms. (Contributed by NM, 10-May-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equs4 | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2390 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exintr 1893 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
3 | 1, 2 | mpi 20 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: equsex 2429 equs45f 2471 equs5 2472 sb1OLD 2496 sb2ALT 2563 bj-sbsb 34275 |
Copyright terms: Public domain | W3C validator |