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Theorem cbvalw 1701
 Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypotheses
Ref Expression
cbvalw.1 (xφyxφ)
cbvalw.2 ψx ¬ ψ)
cbvalw.3 (yψxyψ)
cbvalw.4 φy ¬ φ)
cbvalw.5 (x = y → (φψ))
Assertion
Ref Expression
cbvalw (xφyψ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbvalw
StepHypRef Expression
1 cbvalw.1 . . 3 (xφyxφ)
2 cbvalw.2 . . 3 ψx ¬ ψ)
3 cbvalw.5 . . . 4 (x = y → (φψ))
43biimpd 198 . . 3 (x = y → (φψ))
51, 2, 4cbvaliw 1673 . 2 (xφyψ)
6 cbvalw.3 . . 3 (yψxyψ)
7 cbvalw.4 . . 3 φy ¬ φ)
83biimprd 214 . . . 4 (x = y → (ψφ))
98equcoms 1681 . . 3 (y = x → (ψφ))
106, 7, 9cbvaliw 1673 . 2 (yψxφ)
115, 10impbii 180 1 (xφyψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  hbn1fw  1705
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