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Theorem sbralie 3332
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 3330 . . . 4 (∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
21sbbii 2069 . . 3 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧𝑦 [𝑧 / 𝑥]𝜑)
3 nfv 2009 . . . 4 𝑦𝑧𝑥 [𝑧 / 𝑥]𝜑
4 raleq 3286 . . . 4 (𝑦 = 𝑥 → (∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑))
53, 4sbie 2499 . . 3 ([𝑥 / 𝑦]∀𝑧𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑)
62, 5bitri 266 . 2 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑧𝑥 [𝑧 / 𝑥]𝜑)
7 cbvralsv 3330 . . 3 (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
8 nfv 2009 . . . . . 6 𝑧𝜑
98sbco2 2506 . . . . 5 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
10 nfv 2009 . . . . . 6 𝑥𝜓
11 sbralie.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1210, 11sbie 2499 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
139, 12bitri 266 . . . 4 ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑𝜓)
1413ralbii 3127 . . 3 (∀𝑦𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
157, 14bitri 266 . 2 (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
166, 15bitri 266 1 ([𝑥 / 𝑦]∀𝑥𝑦 𝜑 ↔ ∀𝑦𝑥 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  [wsb 2062  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060
This theorem is referenced by:  tfinds2  7261
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