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Metamath Proof Explorer

Theorem sb3b

Description: Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 6-Oct-2018) Shorten sb3 . (Revised by Wolf Lammen, 21-Feb-2021) (New usage is discouraged.)

		Ref	Expression
	Assertion	sb3b	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sb4b	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2		equs5	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	1 2	bitr4d	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )