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Mirrors > Home > MPE Home > Th. List > Mathboxes > ichnfimlem2 | Structured version Visualization version GIF version |
Description: Lemma for ichnfimlem3 43672: When substituting successively for two always equal variables, the second substitution has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) |
Ref | Expression |
---|---|
ichnfimlem2 | ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1v 2160 | . . 3 ⊢ Ⅎ𝑥[𝑏 / 𝑥]𝜑 | |
2 | nfa1 2155 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦 | |
3 | drsb1 2535 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑏 / 𝑥]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
4 | 2, 3 | nfbidf 2226 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥[𝑏 / 𝑥]𝜑 ↔ Ⅎ𝑥[𝑏 / 𝑦]𝜑)) |
5 | 1, 4 | mpbii 235 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑏 / 𝑦]𝜑) |
6 | sbft 2270 | . 2 ⊢ (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) | |
7 | 5, 6 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 Ⅎwnf 1784 [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-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: ichnfimlem3 43672 |
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