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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 φ ↔ [y / x]y x 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 2846 . . 3 (y x ψz x [z / y]ψ)
21sbbii 1653 . 2 ([y / x]y x ψ ↔ [y / x]z x [z / y]ψ)
3 nfv 1619 . . 3 xz y [z / y]ψ
4 raleq 2807 . . 3 (x = y → (z x [z / y]ψz y [z / y]ψ))
53, 4sbie 2038 . 2 ([y / x]z x [z / y]ψz y [z / y]ψ)
6 cbvralsv 2846 . . 3 (z y [z / y]ψx y [x / z][z / y]ψ)
7 nfv 1619 . . . . . 6 zψ
87sbco2 2086 . . . . 5 ([x / z][z / y]ψ ↔ [x / y]ψ)
9 nfv 1619 . . . . . 6 yφ
10 sbralie.1 . . . . . . . 8 (x = y → (φψ))
1110bicomd 192 . . . . . . 7 (x = y → (ψφ))
1211equcoms 1681 . . . . . 6 (y = x → (ψφ))
139, 12sbie 2038 . . . . 5 ([x / y]ψφ)
148, 13bitri 240 . . . 4 ([x / z][z / y]ψφ)
1514ralbii 2638 . . 3 (x y [x / z][z / y]ψx y φ)
166, 15bitri 240 . 2 (z y [z / y]ψx y φ)
172, 5, 163bitrri 263 1 (x y φ ↔ [y / x]y 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-mp 5  ax-1 6  ax-2 7  ax-3 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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