Theorem aev2 2013

Description: A version of aev 2007 with two universal quantifiers in the consequent, and a generalization of hbaevg 2008. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 2006, aev 2007, aev2 2013).

Using aev 2007 and alrimiv 1887 (as in aev2ALT 2014), 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 28032. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1759, ax-1 6-- ax-13 2302 (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 2008	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑤 𝑤 = 𝑠)
2		aev 2007	. 2 ⊢ (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
3	1, 2	sylg 1786	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744

This theorem is referenced by: (None)