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Mirrors > Home > MPE Home > Th. List > equsal | 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 2365. See equsalvw 1999 and equsalv 2250 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex 2411. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equsal.1 | ⊢ Ⅎ𝑥𝜓 |
equsal.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsal | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsal.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.23 2196 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
3 | equsal.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | pm5.74i 271 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
5 | 4 | albii 1813 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
6 | ax6e 2376 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
7 | 6 | a1bi 362 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
8 | 2, 5, 7 | 3bitr4i 303 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-12 2163 ax-13 2365 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 |
This theorem is referenced by: equsex 2411 equsalh 2413 dvelimf 2441 sb6x 2457 sb6rf 2461 bj-sbievv 36234 |
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