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Mirrors > Home > MPE Home > Th. List > sbbibvv | Structured version Visualization version GIF version |
Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.) |
Ref | Expression |
---|---|
sbbibvv | ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1915 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | sbbib 2380 | 1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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