| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > spimvALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of spimv 2395. Note that it requires only ax-1 6 through ax-5 1910 together with ax6e 2388. Currently, proofs derive from ax6v 1968, but if ax-6 1967 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2141. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| spimv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
| Ref | Expression |
|---|---|
| spimvALT | ⊢ (∀𝑥𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax6e 2388 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
| 2 | spimv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
| 3 | 1, 2 | eximii 1837 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
| 4 | 3 | 19.36iv 1946 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| 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-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |