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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ssblem2 | Structured version Visualization version GIF version |
Description: The converse may not be provable without ax-11 2021. (Contributed by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
bj-ssblem2 | ⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ1 1939 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑡 ↔ 𝑧 = 𝑡)) | |
2 | equequ2 1940 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) | |
3 | 2 | imbi1d 330 | . . 3 ⊢ (𝑦 = 𝑧 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜑))) |
4 | 1, 3 | imbi12d 333 | . 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑧 = 𝑡 → (𝑥 = 𝑧 → 𝜑)))) |
5 | 4 | alcomiw 1958 | 1 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: bj-ssb1a 31821 |
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