Theorem frege55lem1b 44698

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)

Assertion

Ref	Expression
frege55lem1b	⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧))

Distinct variable group:   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem frege55lem1b

Step	Hyp	Ref	Expression
1		equsb3 2141	. . 3 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)
2	1	biimpi 219	. 2 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 → 𝑥 = 𝑧)
3	2	imim2i 17	1 ⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧))

This proof depends on syntax axioms:   → wi 4  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by: (None)