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Mirrors > Home > ILE Home > Th. List > ax11i | GIF version |
Description: Inference that has ax-11 1493 (without ∀𝑦) as its conclusion and does not require ax-10 1492, ax-11 1493, or ax12 1499 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) |
Ref | Expression |
---|---|
ax11i.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
ax11i.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
ax11i | ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax11i.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | ax11i.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 1 | biimprcd 159 | . . 3 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝜑)) |
4 | 2, 3 | alrimih 1456 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 1, 4 | syl6bi 162 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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