Theorem ax7v 2004

Description: Weakened version of ax-7 2003, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2003, and it should be referenced only by its two weakened versions ax7v1 2005 and ax7v2 2006, from which ax-7 2003 is then rederived as ax7 2011, which shows that either ax7v 2004 or the conjunction of ax7v1 2005 and ax7v2 2006 is sufficient.

In ax7v 2004, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2004 "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 2009 and equid 2007 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2011 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 2003	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 2003

This theorem is referenced by:  ax7v1  2005  ax7v2  2006