Theorem ax7v 2009

Description: Weakened version of ax-7 2008, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2008, and it should be referenced only by its two weakened versions ax7v1 2010 and ax7v2 2011, from which ax-7 2008 is then rederived as ax7 2016, which shows that either ax7v 2009 or the conjunction of ax7v1 2010 and ax7v2 2011 is sufficient.

In ax7v 2009, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2009 "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 2014 and equid 2012 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2016 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 2008	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 2008

This theorem is referenced by:  ax7v1  2010  ax7v2  2011