Metamath Proof Explorer

Theorem 2sb6rfOLD

Description: Obsolete version of 2sb6rf as of 13-Apr-2023. (Contributed by NM, 3-Feb-2005) (Revised by Mario Carneiro, 6-Oct-2016) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses 2sb5rf.1 ⊢ Ⅎ z φ
2sb5rf.2 ⊢ Ⅎ w φ
Assertion 2sb6rfOLD ⊢ φ ↔ ∀ z ∀ w z = x ∧ w = y → z x w y φ

Proof

Step Hyp Ref Expression
1 2sb5rf.1 ⊢ Ⅎ z φ
2 2sb5rf.2 ⊢ Ⅎ w φ
3 sbequ12r ⊢ z = x → z x w y φ ↔ w y φ
4 sbequ12r ⊢ w = y → w y φ ↔ φ
5 3 4 sylan9bb ⊢ z = x ∧ w = y → z x w y φ ↔ φ
6 5 pm5.74i ⊢ z = x ∧ w = y → z x w y φ ↔ z = x ∧ w = y → φ
7 6 2albii ⊢ ∀ z ∀ w z = x ∧ w = y → z x w y φ ↔ ∀ z ∀ w z = x ∧ w = y → φ
8 2 19.23 ⊢ ∀ w z = x ∧ w = y → φ ↔ ∃ w z = x ∧ w = y → φ
9 8 albii ⊢ ∀ z ∀ w z = x ∧ w = y → φ ↔ ∀ z ∃ w z = x ∧ w = y → φ
10 1 19.23 ⊢ ∀ z ∃ w z = x ∧ w = y → φ ↔ ∃ z ∃ w z = x ∧ w = y → φ
11 9 10 bitri ⊢ ∀ z ∀ w z = x ∧ w = y → φ ↔ ∃ z ∃ w z = x ∧ w = y → φ
12 2ax6e ⊢ ∃ z ∃ w z = x ∧ w = y
13 pm5.5 ⊢ ∃ z ∃ w z = x ∧ w = y → ∃ z ∃ w z = x ∧ w = y → φ ↔ φ
14 12 13 ax-mp ⊢ ∃ z ∃ w z = x ∧ w = y → φ ↔ φ
15 7 11 14 3bitrri ⊢ φ ↔ ∀ z ∀ w z = x ∧ w = y → z x w y φ