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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax9 | Structured version Visualization version GIF version |
Description: Proof of ax-9 2121 from Tarski's FOL=, sp 2180, dfcleq 2792 and ax-ext 2770 (with two extra disjoint variable conditions on 𝑥, 𝑧 and 𝑦, 𝑧). See ax9ALT 2794 for a more general version, proved using also ax-8 2113. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-ax9 | ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2792 | . 2 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
2 | biimp 218 | . . 3 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | |
3 | 2 | sps 2182 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
4 | 1, 3 | sylbi 220 | 1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 |
This theorem is referenced by: (None) |
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