Theorem aev2 2090

Description: A version of aev 2089 with two universal quantifiers in the consequent. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 2088, aev 2089, aev2 2090).

Using aev 2089 and alrimiv 1957, one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 → PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors →, ↔, ∧, ∨, ⊤, =, ∀, ∃, ∃*, ∃!, Ⅎ. An example is given by aevdemo 30789. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1825, ax-1 6-- ax-13 2404 (as the one-element universe shows), so for instance (∀𝑥𝑥 = 𝑦 → ⊥), DV (𝑥, 𝑦) is not provable from these axioms alone (indeed, dtru 5420 uses non-logical axioms as well). (Contributed by BJ, 23-Mar-2021.)

Assertion

Ref	Expression
aev2	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣)

Distinct variable group:   𝑥,𝑦

Proof of Theorem aev2

Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 2089	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠)
2		aev 2089	. . 3 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
3	2	alrimiv 1957	. 2 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑧∀𝑡 𝑢 = 𝑣)
4	1, 3	syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  hbaev  2091