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Mirrors > Home > MPE Home > Th. List > alxfr | Structured version Visualization version GIF version |
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.) |
Ref | Expression |
---|---|
alxfr.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
alxfr | ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alxfr.1 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | spcgv 3442 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)) |
3 | 2 | com12 32 | . . . . 5 ⊢ (∀𝑥𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
4 | 3 | alimdv 1996 | . . . 4 ⊢ (∀𝑥𝜑 → (∀𝑦 𝐴 ∈ 𝐵 → ∀𝑦𝜓)) |
5 | 4 | com12 32 | . . 3 ⊢ (∀𝑦 𝐴 ∈ 𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓)) |
6 | 5 | adantr 466 | . 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓)) |
7 | nfa1 2183 | . . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
8 | nfv 1994 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
9 | sp 2206 | . . . . . . 7 ⊢ (∀𝑦𝜓 → 𝜓) | |
10 | 9, 1 | syl5ibrcom 237 | . . . . . 6 ⊢ (∀𝑦𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 7, 8, 10 | exlimd 2242 | . . . . 5 ⊢ (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴 → 𝜑)) |
12 | 11 | alimdv 1996 | . . . 4 ⊢ (∀𝑦𝜓 → (∀𝑥∃𝑦 𝑥 = 𝐴 → ∀𝑥𝜑)) |
13 | 12 | com12 32 | . . 3 ⊢ (∀𝑥∃𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑)) |
14 | 13 | adantl 467 | . 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑)) |
15 | 6, 14 | impbid 202 | 1 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∀wal 1628 = wceq 1630 ∃wex 1851 ∈ wcel 2144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-v 3351 |
This theorem is referenced by: (None) |
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