Theorem aev2 2079

Description: A version of aev 2078 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 2077, aev 2078, aev2 2079).

Using aev 2078 and alrimiv 1946, 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 30608. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1814, ax-1 6-- ax-13 2402 (as the one-element universe shows), so for instance (∀𝑥𝑥 = 𝑦 → ⊥), DV (𝑥, 𝑦) is not provable from these axioms alone (indeed, dtru 5403 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 2078	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠)
2		aev 2078	. . 3 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
3	2	alrimiv 1946	. 2 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑧∀𝑡 𝑢 = 𝑣)
4	1, 3	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  hbaev  2080