This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ) )