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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 481 | . . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → 𝜓) |
3 | bnj1465.2 | . . . . 5 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | bnj1465.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | bnj1464 32015 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
6 | 5 | adantl 482 | . . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
7 | 2, 6 | mpbird 258 | . 2 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜑) |
8 | 7 | spesbcd 3863 | 1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-sbc 3770 |
This theorem is referenced by: bnj1463 32224 |
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