Theorem ax7v 1966

Description: Weakened version of ax-7 1965, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 1965, and it should be referenced only by its two weakened versions ax7v1 1967 and ax7v2 1968, from which ax-7 1965 is then rederived as ax7 1973, which shows that either ax7v 1966 or the conjunction of ax7v1 1967 and ax7v2 1968 is sufficient.

In ax7v 1966, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 1966 "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 1971 and equid 1969 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 1973 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 1965	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 1965

This theorem is referenced by:  ax7v1  1967  ax7v2  1968