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Metamath Proof Explorer

Theorem sbc5ALT

Description: Alternate proof of sbc5 . This proof helps show how clelab works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg and isset . (Contributed by NM, 23-Aug-1993) (Revised by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion sbc5ALT ⊢ [˙A / x]˙ φ ↔ ∃ x x = A ∧ φ

Proof

Step Hyp Ref Expression
1 sbcex ⊢ [˙A / x]˙ φ → A ∈ V
2 exsimpl ⊢ ∃ x x = A ∧ φ → ∃ x x = A
3 isset ⊢ A ∈ V ↔ ∃ x x = A
4 2 3 sylibr ⊢ ∃ x x = A ∧ φ → A ∈ V
5 dfsbcq2 ⊢ y = A → y x φ ↔ [˙A / x]˙ φ
6 eqeq2 ⊢ y = A → x = y ↔ x = A
7 6 anbi1d ⊢ y = A → x = y ∧ φ ↔ x = A ∧ φ
8 7 exbidv ⊢ y = A → ∃ x x = y ∧ φ ↔ ∃ x x = A ∧ φ
9 sb5 ⊢ y x φ ↔ ∃ x x = y ∧ φ
10 5 8 9 vtoclbg ⊢ A ∈ V → [˙A / x]˙ φ ↔ ∃ x x = A ∧ φ
11 1 4 10 pm5.21nii ⊢ [˙A / x]˙ φ ↔ ∃ x x = A ∧ φ