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Theorem sbralie 2665
Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
Hypothesis
Ref Expression
sbralie.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
sbralie (∀𝑥𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦𝑥 𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem sbralie
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2663 . . 3 (∀𝑦𝑥 𝜓 ↔ ∀𝑧𝑥 [𝑧 / 𝑦]𝜓)
21sbbii 1738 . 2 ([𝑦 / 𝑥]∀𝑦𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧𝑥 [𝑧 / 𝑦]𝜓)
3 nfv 1508 . . 3 𝑥𝑧𝑦 [𝑧 / 𝑦]𝜓
4 raleq 2624 . . 3 (𝑥 = 𝑦 → (∀𝑧𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧𝑦 [𝑧 / 𝑦]𝜓))
53, 4sbie 1764 . 2 ([𝑦 / 𝑥]∀𝑧𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧𝑦 [𝑧 / 𝑦]𝜓)
6 cbvralsv 2663 . . 3 (∀𝑧𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓)
7 nfv 1508 . . . . . 6 𝑧𝜓
87sbco2 1936 . . . . 5 ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓)
9 nfv 1508 . . . . . 6 𝑦𝜑
10 sbralie.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1110bicomd 140 . . . . . . 7 (𝑥 = 𝑦 → (𝜓𝜑))
1211equcoms 1684 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
139, 12sbie 1764 . . . . 5 ([𝑥 / 𝑦]𝜓𝜑)
148, 13bitri 183 . . . 4 ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓𝜑)
1514ralbii 2439 . . 3 (∀𝑥𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥𝑦 𝜑)
166, 15bitri 183 . 2 (∀𝑧𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥𝑦 𝜑)
172, 5, 163bitrri 206 1 (∀𝑥𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦𝑥 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  [wsb 1735  wral 2414
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419
This theorem is referenced by: (None)
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