Theorem cbvreuvwOLD 3414

Description: Obsolete version of cbvreuvw 3401 as of 30-Sep-2024. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvreuvwOLD.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvreuvwOLD	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuvwOLD

Step	Hyp	Ref	Expression
1		nfv 1918	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1918	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvreuvwOLD.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvreuw 3407	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∃!wreu 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-mo 2535  df-eu 2564  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378

This theorem is referenced by: (None)