Theorem ax7v 2005

Description: Weakened version of ax-7 2004, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2004, and it should be referenced only by its two weakened versions ax7v1 2006 and ax7v2 2007, from which ax-7 2004 is then rederived as ax7 2012, which shows that either ax7v 2005 or the conjunction of ax7v1 2006 and ax7v2 2007 is sufficient.

In ax7v 2005, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2005 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 2010 and equid 2008 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2012 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax7v	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax7v

Step	Hyp	Ref	Expression
1		ax-7 2004	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 2004

This theorem is referenced by:  ax7v1  2006  ax7v2  2007