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Mirrors > Home > MPE Home > Th. List > equsexh | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. See equsexhv 2300 for a version with a disjoint variable condition which does not require ax-13 2390. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equsalh.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
equsalh.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsexh | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalh.1 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | nf5i 2150 | . 2 ⊢ Ⅎ𝑥𝜓 |
3 | equsalh.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | equsex 2440 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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