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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax12ssb | Structured version Visualization version GIF version |
Description: The axiom bj-ax12 31823 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) |
Ref | Expression |
---|---|
bj-ax12ssb | ⊢ [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ax12 31823 | . . 3 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | |
2 | bj-ssb1 31822 | . . . . . 6 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | |
3 | 2 | imbi2i 325 | . . . . 5 ⊢ ((𝜑 → [𝑡/𝑥]b𝜑) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
4 | 3 | imbi2i 325 | . . . 4 ⊢ ((𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
5 | 4 | albii 1737 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
6 | 1, 5 | mpbir 220 | . 2 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) |
7 | bj-ssb1 31822 | . 2 ⊢ ([𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))) | |
8 | 6, 7 | mpbir 220 | 1 ⊢ [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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-11 2021 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: (None) |
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