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Mirrors > Home > MPE Home > Th. List > ceqsalt | Structured version Visualization version GIF version |
Description: Closed theorem version of ceqsalg 3507. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
ceqsalt | ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 214 | . . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | imim3i 64 | . . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))) |
3 | 2 | al2imi 1816 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → ∀𝑥(𝑥 = 𝐴 → 𝜓))) |
4 | elisset 2814 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
5 | 19.23t 2202 | . . . . . . 7 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))) | |
6 | 5 | biimpd 228 | . . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (∃𝑥 𝑥 = 𝐴 → 𝜓))) |
7 | 4, 6 | syl7 74 | . . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (𝐴 ∈ 𝑉 → 𝜓))) |
8 | 3, 7 | sylan9r 508 | . . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝐴 ∈ 𝑉 → 𝜓))) |
9 | 8 | com23 86 | . . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))) |
10 | 9 | 3impia 1116 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓)) |
11 | ceqsal1t 3504 | . . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))) | |
12 | 11 | 3adant3 1131 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))) |
13 | 10, 12 | impbid 211 | 1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2105 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1088 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-clel 2809 |
This theorem is referenced by: ceqsralt 3506 ceqsalg 3507 vtoclegft 3574 |
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