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Mirrors > Home > MPE Home > Th. List > equsal | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. (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.) |
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 2067 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
3 | equsal.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | pm5.74i 259 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
5 | 4 | albii 1737 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
6 | ax6e 2238 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
7 | 6 | a1bi 351 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
8 | 2, 5, 7 | 3bitr4i 291 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: equsalh 2280 equsex 2281 dvelimf 2322 sb6x 2372 sb6rf 2411 |
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