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Theorem ralxfrd 4144
Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfrd.1 ((φ y 𝐶) → A B)
ralxfrd.2 ((φ x B) → y 𝐶 x = A)
ralxfrd.3 ((φ x = A) → (ψχ))
Assertion
Ref Expression
ralxfrd (φ → (x B ψy 𝐶 χ))
Distinct variable groups:   x,A   x,y,B   x,𝐶   χ,x   φ,x,y   ψ,y
Allowed substitution hints:   ψ(x)   χ(y)   A(y)   𝐶(y)

Proof of Theorem ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4 ((φ y 𝐶) → A B)
2 ralxfrd.3 . . . . 5 ((φ x = A) → (ψχ))
32adantlr 449 . . . 4 (((φ y 𝐶) x = A) → (ψχ))
41, 3rspcdv 2636 . . 3 ((φ y 𝐶) → (x B ψχ))
54ralrimdva 2377 . 2 (φ → (x B ψy 𝐶 χ))
6 ralxfrd.2 . . . 4 ((φ x B) → y 𝐶 x = A)
7 r19.29 2428 . . . . 5 ((y 𝐶 χ y 𝐶 x = A) → y 𝐶 (χ x = A))
82biimprd 147 . . . . . . . . 9 ((φ x = A) → (χψ))
98expimpd 345 . . . . . . . 8 (φ → ((x = A χ) → ψ))
109ancomsd 256 . . . . . . 7 (φ → ((χ x = A) → ψ))
1110ad2antrr 460 . . . . . 6 (((φ x B) y 𝐶) → ((χ x = A) → ψ))
1211rexlimdva 2411 . . . . 5 ((φ x B) → (y 𝐶 (χ x = A) → ψ))
137, 12syl5 28 . . . 4 ((φ x B) → ((y 𝐶 χ y 𝐶 x = A) → ψ))
146, 13mpan2d 406 . . 3 ((φ x B) → (y 𝐶 χψ))
1514ralrimdva 2377 . 2 (φ → (y 𝐶 χx B ψ))
165, 15impbid 120 1 (φ → (x B ψy 𝐶 χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wral 2284  wrex 2285
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537
This theorem is referenced by:  ralxfr2d  4146  ralxfr  4148
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