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Mirrors > Home > MPE Home > Th. List > spimvALT | Structured version Visualization version GIF version |
Description: Alternate proof of spimv 2393. Note that it requires only ax-1 6 through ax-5 1908 together with ax6e 2386. Currently, proofs derive from ax6v 1966, but if ax-6 1965 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2139. (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 2386 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | spimv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
3 | 1, 2 | eximii 1834 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
4 | 3 | 19.36iv 1944 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: (None) |
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