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Theorem ax11-pm2 34665
Description: Proof of ax-11 2162 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2164 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2162 is used in the proof only through nfal 2326, nfsb 2528, sbal 2167, sb8 2522. See also ax11-pm 34661. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax11-pm2 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11-pm2
Dummy variables 𝑧 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2stdpc4 2080 . . . . . 6 (∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
21gen2 1803 . . . . 5 𝑡𝑧(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
3 nfv 1921 . . . . . . . 8 𝑡𝜑
43nfal 2326 . . . . . . 7 𝑡𝑦𝜑
54nfal 2326 . . . . . 6 𝑡𝑥𝑦𝜑
6 nfv 1921 . . . . . . . 8 𝑧𝜑
76nfal 2326 . . . . . . 7 𝑧𝑦𝜑
87nfal 2326 . . . . . 6 𝑧𝑥𝑦𝜑
95, 82stdpc5 34658 . . . . 5 (∀𝑡𝑧(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑡 / 𝑦]𝜑) → (∀𝑥𝑦𝜑 → ∀𝑡𝑧[𝑧 / 𝑥][𝑡 / 𝑦]𝜑))
102, 9ax-mp 5 . . . 4 (∀𝑥𝑦𝜑 → ∀𝑡𝑧[𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
116nfsbv 2333 . . . . . 6 𝑧[𝑡 / 𝑦]𝜑
1211sb8v 2356 . . . . 5 (∀𝑥[𝑡 / 𝑦]𝜑 ↔ ∀𝑧[𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
1312albii 1826 . . . 4 (∀𝑡𝑥[𝑡 / 𝑦]𝜑 ↔ ∀𝑡𝑧[𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
1410, 13sylibr 237 . . 3 (∀𝑥𝑦𝜑 → ∀𝑡𝑥[𝑡 / 𝑦]𝜑)
15 sbal 2167 . . . 4 ([𝑡 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑡 / 𝑦]𝜑)
1615albii 1826 . . 3 (∀𝑡[𝑡 / 𝑦]∀𝑥𝜑 ↔ ∀𝑡𝑥[𝑡 / 𝑦]𝜑)
1714, 16sylibr 237 . 2 (∀𝑥𝑦𝜑 → ∀𝑡[𝑡 / 𝑦]∀𝑥𝜑)
183nfal 2326 . . 3 𝑡𝑥𝜑
1918sb8v 2356 . 2 (∀𝑦𝑥𝜑 ↔ ∀𝑡[𝑡 / 𝑦]∀𝑥𝜑)
2017, 19sylibr 237 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  [wsb 2074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791  df-sb 2075
This theorem is referenced by: (None)
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