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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ssbequ1 | Structured version Visualization version GIF version |
Description: This uses ax-12 2034 with a direct reference to ax12v 2035. Therefore, compared to bj-ax12 31823, there is a hidden use of sp 2041. Note that with ax-12 2034, it can be proved with dv condition on 𝑥, 𝑡. See sbequ1 2096. (Contributed by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
bj-ssbequ1 | ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtr2 1941 | . . . . . . . 8 ⊢ ((𝑦 = 𝑡 ∧ 𝑥 = 𝑡) → 𝑦 = 𝑥) | |
2 | 1 | equcomd 1933 | . . . . . . 7 ⊢ ((𝑦 = 𝑡 ∧ 𝑥 = 𝑡) → 𝑥 = 𝑦) |
3 | ax12v 2035 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ ((𝑦 = 𝑡 ∧ 𝑥 = 𝑡) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | 4 | expimpd 627 | . . . . 5 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
6 | 5 | com12 32 | . . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
7 | 6 | alrimiv 1842 | . . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
8 | 7 | ex 449 | . 2 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
9 | df-ssb 31809 | . 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
10 | 8, 9 | syl6ibr 241 | 1 ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 [wssb 31808 |
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 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ssb 31809 |
This theorem is referenced by: bj-ssbid1 31836 |
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