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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-equsald | Structured version Visualization version GIF version |
Description: Deduction version of equsal 2279. (Contributed by Wolf Lammen, 27-Jul-2019.) |
Ref | Expression |
---|---|
wl-equsald.1 | ⊢ Ⅎ𝑥𝜑 |
wl-equsald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
wl-equsald.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
wl-equsald | ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-equsald.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
2 | 19.23t 2066 | . . 3 ⊢ (Ⅎ𝑥𝜒 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒))) |
4 | wl-equsald.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
5 | wl-equsald.3 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 5 | pm5.74d 261 | . . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜓) ↔ (𝑥 = 𝑦 → 𝜒))) |
7 | 4, 6 | albid 2077 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜒))) |
8 | ax6e 2238 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
9 | 8 | a1bi 351 | . . 3 ⊢ (𝜒 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒)) |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → (𝜒 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜒))) |
11 | 3, 7, 10 | 3bitr4d 299 | 1 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) |
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: wl-equsal 32505 wl-equsal1t 32506 wl-sb6rft 32509 |
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