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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax12ig | Structured version Visualization version GIF version |
Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 32616. (Contributed by BJ, 19-Dec-2020.) |
Ref | Expression |
---|---|
bj-ax12ig.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bj-ax12ig.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
Ref | Expression |
---|---|
bj-ax12ig | ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ax12ig.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | pm5.32i 669 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) |
3 | bj-ax12ig.2 | . . . . 5 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
4 | 3 | imp 445 | . . . 4 ⊢ ((𝜑 ∧ 𝜒) → ∀𝑥𝜒) |
5 | 1 | biimprcd 240 | . . . 4 ⊢ (𝜒 → (𝜑 → 𝜓)) |
6 | 4, 5 | sylg 1750 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓)) |
7 | 2, 6 | sylbi 207 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
8 | 7 | ex 450 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-an 386 |
This theorem is referenced by: bj-ax12i 32616 |
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