Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfbii2 | Structured version Visualization version GIF version |
Description: Equality deduction for not-freeness. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
Ref | Expression |
---|---|
nfbii2 | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2028 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝜓) | |
2 | sp 2053 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | nfbidf 2092 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 Ⅎwnf 1708 |
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-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-ex 1705 df-nf 1710 |
This theorem is referenced by: (None) |
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