ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralxfrALT Structured version   GIF version

Theorem ralxfrALT 4165
Description: Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4160. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ralxfr.1 (y 𝐶A B)
ralxfr.2 (x By 𝐶 x = A)
ralxfr.3 (x = A → (φψ))
Assertion
Ref Expression
ralxfrALT (x B φy 𝐶 ψ)
Distinct variable groups:   ψ,x   φ,y   x,A   x,y,B   x,𝐶
Allowed substitution hints:   φ(x)   ψ(y)   A(y)   𝐶(y)

Proof of Theorem ralxfrALT
StepHypRef Expression
1 ralxfr.1 . . . . 5 (y 𝐶A B)
2 ralxfr.3 . . . . . 6 (x = A → (φψ))
32rspcv 2646 . . . . 5 (A B → (x B φψ))
41, 3syl 14 . . . 4 (y 𝐶 → (x B φψ))
54com12 27 . . 3 (x B φ → (y 𝐶ψ))
65ralrimiv 2385 . 2 (x B φy 𝐶 ψ)
7 ralxfr.2 . . . 4 (x By 𝐶 x = A)
8 nfra1 2349 . . . . 5 yy 𝐶 ψ
9 nfv 1418 . . . . 5 yφ
10 rsp 2363 . . . . . 6 (y 𝐶 ψ → (y 𝐶ψ))
112biimprcd 149 . . . . . 6 (ψ → (x = Aφ))
1210, 11syl6 29 . . . . 5 (y 𝐶 ψ → (y 𝐶 → (x = Aφ)))
138, 9, 12rexlimd 2424 . . . 4 (y 𝐶 ψ → (y 𝐶 x = Aφ))
147, 13syl5 28 . . 3 (y 𝐶 ψ → (x Bφ))
1514ralrimiv 2385 . 2 (y 𝐶 ψx B φ)
166, 15impbii 117 1 (x B φy 𝐶 ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator