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Theorem sbralie 3447
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 cbvralsvw 3443 . . 3 (∀𝑦𝑥 𝜓 ↔ ∀𝑧𝑥 [𝑧 / 𝑦]𝜓)
21sbbii 2081 . 2 ([𝑦 / 𝑥]∀𝑦𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧𝑥 [𝑧 / 𝑦]𝜓)
3 raleq 3389 . . 3 (𝑥 = 𝑦 → (∀𝑧𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧𝑦 [𝑧 / 𝑦]𝜓))
43sbievw 2103 . 2 ([𝑦 / 𝑥]∀𝑧𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧𝑦 [𝑧 / 𝑦]𝜓)
5 cbvralsvw 3443 . . 3 (∀𝑧𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓)
6 sbco2vv 2108 . . . . 5 ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓)
7 sbralie.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
87bicomd 225 . . . . . . 7 (𝑥 = 𝑦 → (𝜓𝜑))
98equcoms 2027 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
109sbievw 2103 . . . . 5 ([𝑥 / 𝑦]𝜓𝜑)
116, 10bitri 277 . . . 4 ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓𝜑)
1211ralbii 3152 . . 3 (∀𝑥𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥𝑦 𝜑)
135, 12bitri 277 . 2 (∀𝑧𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥𝑦 𝜑)
142, 4, 133bitrri 300 1 (∀𝑥𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦𝑥 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  [wsb 2069  wral 3125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130
This theorem is referenced by:  tfinds2  7552
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