Theorem cbveu 1018

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbveu.1	⊢ (φ → ∀yφ)
cbveu.2	⊢ (ψ → ∀xψ)
cbveu.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbveu	⊢ (∃!xφ ↔ ∃!yψ)

Distinct variable group(s):   x,y

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 ⊢ (φ → ∀yφ)
2	1	sb8eu 1017	. 2 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
3		cbveu.2	. . . 4 ⊢ (ψ → ∀xψ)
4		cbveu.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
5	3, 4	sbie 904	. . 3 ⊢ ([y / x]φ ↔ ψ)
6	5	bieu 1014	. 2 ⊢ (∃!y[y / x]φ ↔ ∃!yψ)
7	2, 6	bitr 151	1 ⊢ (∃!xφ ↔ ∃!yψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797  [wsb 852  ∃!weu 1007

This theorem is referenced by:  cbvmo 1034  cbvreuv 1335  euuni 1954  fnopabg 2745  tz6.12f 2844

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009

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