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Metamath Proof Explorer

Theorem cbv2h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 11-May-1993) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	cbv2h.1	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
		cbv2h.2	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
		cbv2h.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
	Assertion	cbv2h	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbv2h.1	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
2		cbv2h.2	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
3		cbv2h.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4		biimp	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜓 → 𝜒 ) )
5	3 4	syl6	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
6	1 2 5	cbv1h	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )
7		equcomi	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
8		biimpr	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜒 → 𝜓 ) )
9	7 3 8	syl56	⊢ ( 𝜑 → ( 𝑦 = 𝑥 → ( 𝜒 → 𝜓 ) ) )
10	2 1 9	cbv1h	⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
11	10	alcoms	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
12	6 11	impbid	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )