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Theorem ax11-pm2 33159
Description: Proof of ax-11 2190 from the standard axioms of predicate calculus, similar to PM's proof of alcom 2193 (PM*11.2). This proof requires that 𝑥 and 𝑦 be distinct. Axiom ax-11 2190 is used in the proof only through nfal 2317, nfsb 2590, sbal 2610, sb8 2571. See also ax11-pm 33155. (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 2500 . . . . . 6 (∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
21gen2 1871 . . . . 5 𝑡𝑧(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
3 nfv 1995 . . . . . . . 8 𝑡𝜑
43nfal 2317 . . . . . . 7 𝑡𝑦𝜑
54nfal 2317 . . . . . 6 𝑡𝑥𝑦𝜑
6 nfv 1995 . . . . . . . 8 𝑧𝜑
76nfal 2317 . . . . . . 7 𝑧𝑦𝜑
87nfal 2317 . . . . . 6 𝑧𝑥𝑦𝜑
95, 82stdpc5 33152 . . . . 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)
116nfsb 2590 . . . . . 6 𝑧[𝑡 / 𝑦]𝜑
1211sb8 2571 . . . . 5 (∀𝑥[𝑡 / 𝑦]𝜑 ↔ ∀𝑧[𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
1312albii 1895 . . . 4 (∀𝑡𝑥[𝑡 / 𝑦]𝜑 ↔ ∀𝑡𝑧[𝑧 / 𝑥][𝑡 / 𝑦]𝜑)
1410, 13sylibr 224 . . 3 (∀𝑥𝑦𝜑 → ∀𝑡𝑥[𝑡 / 𝑦]𝜑)
15 sbal 2610 . . . 4 ([𝑡 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑡 / 𝑦]𝜑)
1615albii 1895 . . 3 (∀𝑡[𝑡 / 𝑦]∀𝑥𝜑 ↔ ∀𝑡𝑥[𝑡 / 𝑦]𝜑)
1714, 16sylibr 224 . 2 (∀𝑥𝑦𝜑 → ∀𝑡[𝑡 / 𝑦]∀𝑥𝜑)
183nfal 2317 . . 3 𝑡𝑥𝜑
1918sb8 2571 . 2 (∀𝑦𝑥𝜑 ↔ ∀𝑡[𝑡 / 𝑦]∀𝑥𝜑)
2017, 19sylibr 224 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by: (None)
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