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Theorem sbralie 2540
 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 2538 . . . 4 (x y φz y [z / x]φ)
21sbbii 1645 . . 3 ([x / y]x y φ ↔ [x / y]z y [z / x]φ)
3 nfv 1418 . . . 4 yz x [z / x]φ
4 raleq 2499 . . . 4 (y = x → (z y [z / x]φz x [z / x]φ))
53, 4sbie 1671 . . 3 ([x / y]z y [z / x]φz x [z / x]φ)
62, 5bitri 173 . 2 ([x / y]x y φz x [z / x]φ)
7 cbvralsv 2538 . . 3 (z x [z / x]φy x [y / z][z / x]φ)
8 nfv 1418 . . . . . 6 zφ
98sbco2 1836 . . . . 5 ([y / z][z / x]φ ↔ [y / x]φ)
10 nfv 1418 . . . . . 6 xψ
11 sbralie.1 . . . . . 6 (x = y → (φψ))
1210, 11sbie 1671 . . . . 5 ([y / x]φψ)
139, 12bitri 173 . . . 4 ([y / z][z / x]φψ)
1413ralbii 2324 . . 3 (y x [y / z][z / x]φy x ψ)
157, 14bitri 173 . 2 (z x [z / x]φy x ψ)
166, 15bitri 173 1 ([x / y]x y φy x ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  [wsb 1642  ∀wral 2300 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305 This theorem is referenced by: (None)
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