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Theorem sbralie 3177
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 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑦𝑥 𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem sbralie
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 3175 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
21sbbii 1889 . . 3 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
3 nfv 1845 . . . 4 𝑦𝑧𝑥 [𝑧 / 𝑥]𝜑
4 raleq 3132 . . . 4 (𝑦 = 𝑥 → (∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑))
53, 4sbie 2412 . . 3 ([𝑥 / 𝑦]∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑)
62, 5bitri 264 . 2 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑)
7 cbvralsv 3175 . . 3 (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
8 nfv 1845 . . . . . 6 𝑧𝜑
98sbco2 2419 . . . . 5 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
10 nfv 1845 . . . . . 6 𝑥𝜓
11 sbralie.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1210, 11sbie 2412 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
139, 12bitri 264 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑𝜓)
1413ralbii 2979 . . 3 (∀𝑦𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
157, 14bitri 264 . 2 (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
166, 15bitri 264 1 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑦𝑥 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1882  wral 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917
This theorem is referenced by:  tfinds2  7011
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