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Mirrors > Home > MPE Home > Th. List > hblem | Structured version Visualization version GIF version |
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2786. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
hblem.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Ref | Expression |
---|---|
hblem | ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hblem.1 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
2 | 1 | hbsb 2469 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴) |
3 | clelsb3 2758 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | albii 1787 | . 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴) |
5 | 2, 3, 4 | 3imtr3i 280 | 1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1521 [wsb 1937 ∈ wcel 2030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-cleq 2644 df-clel 2647 |
This theorem is referenced by: nfcrii 2786 bnj1311 31218 |
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