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Theorem sbralie 2848
 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]x y φy x 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 2846 . . . 4 (x y φz y [z / x]φ)
21sbbii 1653 . . 3 ([x / y]x y φ ↔ [x / y]z y [z / x]φ)
3 nfv 1619 . . . 4 yz x [z / x]φ
4 raleq 2807 . . . 4 (y = x → (z y [z / x]φz x [z / x]φ))
53, 4sbie 2038 . . 3 ([x / y]z y [z / x]φz x [z / x]φ)
62, 5bitri 240 . 2 ([x / y]x y φz x [z / x]φ)
7 cbvralsv 2846 . . 3 (z x [z / x]φy x [y / z][z / x]φ)
8 nfv 1619 . . . . . 6 zφ
98sbco2 2086 . . . . 5 ([y / z][z / x]φ ↔ [y / x]φ)
10 nfv 1619 . . . . . 6 xψ
11 sbralie.1 . . . . . 6 (x = y → (φψ))
1210, 11sbie 2038 . . . . 5 ([y / x]φψ)
139, 12bitri 240 . . . 4 ([y / z][z / x]φψ)
1413ralbii 2638 . . 3 (y x [y / z][z / x]φy x ψ)
157, 14bitri 240 . 2 (z x [z / x]φy x ψ)
166, 15bitri 240 1 ([x / y]x y φy x ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by: (None)
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