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Mirrors > Home > MPE Home > Th. List > 19.21 | Structured version Visualization version GIF version |
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 1855 for a version requiring fewer axioms. See also 19.21h 2107. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
Ref | Expression |
---|---|
19.21.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.21 | ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 19.21t 2061 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: stdpc5 2063 19.21-2 2065 19.32 2088 nf6 2103 19.21h 2107 19.12vv 2168 cbv1 2255 axc14 2360 r2alf 2922 19.12b 30951 bj-biexal2 31884 bj-bialal 31886 bj-cbv1v 31916 wl-dral1d 32497 mpt2bi123f 33141 |
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