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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 1908 for a version requiring fewer axioms. See also 19.21h 2159. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1750 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 2111 | . 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 196 ∀wal 1521 Ⅎwnf 1748 |
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-12 2087 |
This theorem depends on definitions: df-bi 197 df-ex 1745 df-nf 1750 |
This theorem is referenced by: stdpc5 2114 19.21-2 2116 19.32 2139 nf6 2155 19.21h 2159 19.12vv 2216 cbv1 2303 axc14 2400 r2alf 2967 19.12b 31831 bj-biexal2 32822 bj-bialal 32824 bj-cbv1v 32854 wl-dral1d 33448 mpt2bi123f 34101 |
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