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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1465 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1465.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
bnj1465.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
bnj1465.3 | ⊢ (𝜒 → 𝜓) |
Ref | Expression |
---|---|
bnj1465 | ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1465.3 | . . . 4 ⊢ (𝜒 → 𝜓) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → 𝜓) |
3 | bnj1465.2 | . . . . 5 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | bnj1465.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | bnj1464 34384 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
7 | 2, 6 | mpbird 257 | . 2 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜑) |
8 | 7 | spesbcd 3872 | 1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 [wsbc 3772 |
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-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-v 3470 df-sbc 3773 |
This theorem is referenced by: bnj1463 34595 |
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