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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax11-pm | Structured version Visualization version GIF version |
Description: Proof of ax-11 2159 similar to PM's proof of alcom 2161 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 34782. Axiom ax-11 2159 is used in the proof only through nfa2 2175. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ax11-pm | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sp 2184 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 → 𝜑) | |
2 | 1 | gen2 1804 | . 2 ⊢ ∀𝑦∀𝑥(∀𝑥∀𝑦𝜑 → 𝜑) |
3 | nfa2 2175 | . . 3 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑 | |
4 | nfa1 2153 | . . 3 ⊢ Ⅎ𝑥∀𝑥∀𝑦𝜑 | |
5 | 3, 4 | 2stdpc5 34775 | . 2 ⊢ (∀𝑦∀𝑥(∀𝑥∀𝑦𝜑 → 𝜑) → (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2142 ax-11 2159 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: (None) |
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