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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 2757. (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 2441 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴) |
3 | clelsb3 2729 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | albii 1747 | . 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 1481 [wsb 1880 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: nfcrii 2757 bnj1311 31092 |
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