Theorem cbveu 1389

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:   x,y

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 ⊢ (φ → ∀yφ)
2	1	sb8eu 1388	. 2 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
3		cbveu.2	. . . 4 ⊢ (ψ → ∀xψ)
4		cbveu.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
5	3, 4	sbie 1194	. . 3 ⊢ ([y / x]φ ↔ ψ)
6	5	eubii 1385	. 2 ⊢ (∃!y[y / x]φ ↔ ∃!yψ)
7	2, 6	bitr 173	1 ⊢ (∃!xφ ↔ ∃!yψ)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952   = wceq 954  ∃!weu 1378

This theorem is referenced by:  cbvmo 1406  cbvreuv 1798  euuni 2876  fnopabg 3607  tz6.12f 3729  climeu 7045

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380

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