Theorem aev2 1973

Description: A version of aev 1970 with two universal quantifiers in the consequent, and a generalization of hbaevg 1971. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 1969, aev 1970, aev2 1973).

Using aev 1970 and alrimiv 1842 (as in aev2ALT 1974), 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 26709. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1713, ax-1 6-- ax-13 2234 (as the one-element universe shows).

(Contributed by BJ, 29-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		hbaevg 1971	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑤 𝑤 = 𝑠)
2		aev 1970	. 2 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
3	1, 2	sylg 1740	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)