Theorem a12stdy3 1372

Description: Part of a study related to ax-12 966. The consequent introduces two new variables. There are no distinct variable restrictions.

Assertion

Ref	Expression
a12stdy3	⊢ (∀z(z = x ⋀ x = y) → ∀v∃y x = w)

Proof of Theorem a12stdy3

Step	Hyp	Ref	Expression
1		a12stdy2 1371	. 2 ⊢ (∀z(z = x ⋀ x = y) → ∀y y = x)
2		hbae 1143	. 2 ⊢ (∀y y = x → ∀v∀y y = x)
3		a12stdy1 1370	. . 3 ⊢ (∀y y = x → ∃y x = w)
4	3	19.20i 990	. 2 ⊢ (∀v∀y y = x → ∀v∃y x = w)
5	1, 2, 4	3syl 20	1 ⊢ (∀z(z = x ⋀ x = y) → ∀v∃y x = w)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954  ∃wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-9 963  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-10o 1138

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

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