Theorem ax7v 2006

Description: Weakened version of ax-7 2005, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2005, and it should be referenced only by its two weakened versions ax7v1 2007 and ax7v2 2008, from which ax-7 2005 is then rederived as ax7 2013, which shows that either ax7v 2006 or the conjunction of ax7v1 2007 and ax7v2 2008 is sufficient.

In ax7v 2006, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2006 "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 2011 and equid 2009 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2013 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 2005	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 2005

This theorem is referenced by:  ax7v1  2007  ax7v2  2008