Theorem pm13.192

Description: Theorem *13.192 in WhiteheadRussell p. 179. (Contributed by Andrew Salmon, 3-Jun-2011) (Revised by NM, 4-Jan-2017)

	Ref	Expression
Assertion	pm13.192	⊢ ( ∃ 𝑦 ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ [ 𝐴 / 𝑦 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		biimpr	⊢ ( ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) → ( 𝑥 = 𝑦 → 𝑥 = 𝐴 ) )
2	1	alimi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝑥 = 𝐴 ) )
3		eqeq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑦 = 𝐴 ) )
4	3	equsalvw	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝑥 = 𝐴 ) ↔ 𝑦 = 𝐴 )
5	2 4	sylib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) → 𝑦 = 𝐴 )
6		eqeq2	⊢ ( 𝐴 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) )
7	6	eqcoms	⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) )
8	7	alrimiv	⊢ ( 𝑦 = 𝐴 → ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) )
9	5 8	impbii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ↔ 𝑦 = 𝐴 )
10	9	anbi1i	⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ ( 𝑦 = 𝐴 ∧ 𝜑 ) )
11	10	exbii	⊢ ( ∃ 𝑦 ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) )
12		sbc5	⊢ ( [ 𝐴 / 𝑦 ] 𝜑 ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝜑 ) )
13	11 12	bitr4i	⊢ ( ∃ 𝑦 ( ∀ 𝑥 ( 𝑥 = 𝐴 ↔ 𝑥 = 𝑦 ) ∧ 𝜑 ) ↔ [ 𝐴 / 𝑦 ] 𝜑 )