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Mirrors > Home > MPE Home > Th. List > equsalv | Structured version Visualization version GIF version |
Description: Version of equsal 2397 with a disjoint variable condition, which does not require ax-13 2346. See equsalvw 1991 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2234. (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalv.nf | ⊢ Ⅎ𝑥𝜓 |
equsalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsalv | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.23 2178 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
3 | equsalv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | pm5.74i 272 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
5 | 4 | albii 1805 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
6 | ax6ev 1953 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
7 | 6 | a1bi 364 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
8 | 2, 5, 7 | 3bitr4i 304 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1523 ∃wex 1765 Ⅎwnf 1769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-12 2143 |
This theorem depends on definitions: df-bi 208 df-ex 1766 df-nf 1770 |
This theorem is referenced by: equsalhw 2267 sbiev 2297 sb6rfv 2335 bj-equsalhv 33681 |
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