Theorem ax7v 2042

Description: Weakened version of ax-7 2041, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2041, and it should be referenced only by its two weakened versions ax7v1 2043 and ax7v2 2044, from which ax-7 2041 is then rederived as ax7 2049, which shows that either ax7v 2042 or the conjunction of ax7v1 2043 and ax7v2 2044 is sufficient.

In ax7v 2042, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2042 "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 2047 and equid 2045 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2049 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 2041	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-7 2041

This theorem is used by:  ax7v1  2043  ax7v2  2044