Theorem ax7v 2010

Description: Weakened version of ax-7 2009, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2009, and it should be referenced only by its two weakened versions ax7v1 2011 and ax7v2 2012, from which ax-7 2009 is then rederived as ax7 2017, which shows that either ax7v 2010 or the conjunction of ax7v1 2011 and ax7v2 2012 is sufficient.

In ax7v 2010, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2010 "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 2015 and equid 2013 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2017 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 2009	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 2009

This theorem is referenced by:  ax7v1  2011  ax7v2  2012