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Mirrors > Home > MPE Home > Th. List > hbim | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
Ref | Expression |
---|---|
hbim.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbim.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
hbim | ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbim.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | hbim.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
4 | 1, 3 | hbim1 2287 | 1 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 |
This theorem is referenced by: axi5r 2691 hbral 3301 |
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